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Zbc  mniversiti?  of  Cbicago 

FOUNDED  BY  JOHN  O.  ROCKEFELLER 


THE  PROBLEM  OF  THE  ANGLE 

BISECTORS 


A  DISSERTATION 

SUBMITTED    TO    THE    FACULTY    OF    THE    OGDEN    GRADUATE    SCHOOL    OF 

SCIENCE  OF  THE  UNIVERSITY  OF  CHICAGO  IN  CANDIDACY 

FOR  THE  DEGREE  OF  DOCTOR  OF  PHILOSOPHY 

(department  of  MATHEMATICS) 


BY 


RICHARD  PHILIP  BAKER 


THE  UNIVERSITY  OF  CHICAGO  PRESS 
CHICAGO,  ILLINOIS 


«ik 


THE  UNIVERSITY  OF  OHIOAGO  PRB88 
OHIOAGO.  ILLINOIS 

Bgents 
THE  BAKER  &  TAYLOR  COMPANY 

HEW   TOBK 

OAMBRIDGB  UNIVERSITY  PRESS 

LONDON    AND    EDINBUBOB 


Zbc  mnipersiti?  of  Cbicago 

POUNDED    BY  JOHN    O.   ROCKEFELLER 


THE  PROBLEM  OF  THE  ANGLE- 
BISECTORS 


A  DISSERTATION 

SUBMITTED    TO    THE    FACULTY    OF    THE    OGDEN    GRADUATE    SCHOOL    OF 

SCIENCE  OF  THE  UNIVERSITY  OF  CHICAGO  IN  CANDIDACY 

FOR  THE  DEGREE  OF  DOCTOR  OF  PHILOSOPHY 

(department  of  MATHEMATICS) 


BY 

RICHARD  PHILIP  BAKER 


THE  UNIVERSITY  OF  CHICAGO  PRESS 
CHICAGO,  ILLINOIS 


Copyright  ign  By 
The  University  of  Chicago 


All  rights  reserved 


Published  March  igii 


Composed  and  Printed  By 

The  University  of  Chicago  Press 

Chicago,  Ilh'nois.  U.S  A. 


TABLE  OF   CONTENTS 


The  History  of  the  Problem 
Method  and  Results 


V 

vi 


I 

1.  The  Internal  Problem 

2.  The  Order  of  the  Problem 

3.  The  Elimination 

4.  The  Parametric  Fields 

5.  Two  Special  Cases    . 

6.  Reality  of  the  Solutions  .. 

7.  The  Elementary  Theory  of  Equations  Applied 

8.  Reality  of  the  Roots  for  Real  Angle-Bisectors 

9.  A  Practical  Method  for  Approximate  Solutions 

10.  Multiple  Points 

1 1 .  The  Transformations 

12.  Finite  Multiple  Points 

13.  The  Discriminant 

14.  The  Monodromie  Group  of  the  Equation  F  (cr,  a,  j3)  =  o 

15.  The  Equation  fofthe-Sides 

16.  The  Monodromie  Group  of  the  Equation  for 'the  Sides 

17.  The  Reduction  of  the  Equation  for  <r  in  the  Case  of  Equal  Bisectors 

18.  The  Reduction  of  the  Equation  for  the  Side  in  the  Case  of  Equal  Bisectors 

19.  The  Surf  ace  F  (»■,  o,  0)  =  o 


II 


1.  The  External  Problem 

2.  The  First  Elimination 

3.  The  Group  of  the  Equation 

4.  The  Second  Elimination  . 

5.  Multiple  Roots 

6.  The  Nodal  Curve      . 

7.  Finite  Multiple  Points 

8.  The  Discriminant 

9.  Interrelations  of  the  Two  Equations 

10.  The  Transformations 

11.  The  Determination  of  the  Sides 

12.  The  Case  of  Equal  Bisectors  (External) 


I 
I 
2 
S 
7 
8 
8 

13 
16 
16 
20 
43 
45 
46 

49 
52 
53 
55 
59 


60 
60 
61 

63 
64 
66 
67 
67 
69 
70 
81 
82 


224663 


IV 


TABLE   OF   CONTENTS 


III 

1.  The  Mixed  Problem 

2.  The  Monodromie  Group 

3.  The  Equation  for  the  Tangent  of  a  Half-Angle 

4.  The  Solution  of  the  Real  Problem 

5.  The  Character  of  the  Solutions 

6.  The  Case  of  Equal  Bisectors   .... 


83 
83 
84 
86 

87 
88 


IV 

1.  The  General  Problem  for  Real  Data 

2.  The  Problem  When  a  Right  Angle  and  Two  Bisectors  Are  Given 

3.  Special  Cases  of  Isosceles  Triangles 

4.  The  Identical  Relations  among  the  Six  Bisectors 

5.  The  Indirect  Proof 

6.  Generalizations  of  the  Problem 


89 
89 
93 
95 
96 

98 


Vita 


99 


THE  HISTORY  OF  THE  PROBLEM 

The  problem  of  constructing  a  triangle  when  the  lengths  of  the  bisectors  of  the  angles  are 
given  has  been  an  outstanding  problem  among  geometers  probably  from  the  time  of  Pascal' 
and  certainly  from  the  time  of  Euler.'^ 

Brocard^  has  summed  up  the  literature,  dealing  almost  entirely  with  special  cases,  of 
which  the  most  extensive  treatment  appears  to  be  the  solution  of  the  problem  when  one  angle 
is  a  right  angle  due  to  Marcus  Baker. ^  This  problem  is  of  the  sixth  order.  Among  many 
special  treatments  that  appear  in  the  smaller  journals  the  fundamental  problem  of  determining 
the  character  of  the  algebraic  irrationality  involved  is  not  mentioned.  As  a  result  apparently 
conflicting  statements  occur  as  to  the  order  of  the  equation  concerned,  this  depending  on  acci- 
dental choice  of  the  parameter  field. 

The  only  paper  dealing  with  the  general  case  where  the  internal  bisector  formulas  are 
used  is  P.  Barbarin's.'  The  case  where  the  external  formulas  are  to  be  used  and  the  case 
where  two  of  the  assigned  bisectors  refer  to  the  same  vertex  is  not  treated  in  general  in  any 
paper  known  to  the  writer. 

Barbarin  proved  that  the  general  internal  problem  could  be  solved  by  the  solution  of  an 
algebraic  equation  of  order  not  greater  than  twelve.  The  irreducibility  and  group  of  the 
equation  are  not  discussed,  and  as  the  equation  itself  is  not  set  out  explicitly  further  reduction 
of  the  order  of  the  problem  is  not  precluded. 

The  method  of  attack  used  by  Barbarin  is  to  solve  first  the  problem  when  an  angle  and 
two  bisectors  are  given,  and  to  use  the  result  as  a  basis  for  attacking  the  general  problem. 
The  necessary  sacrifice  of"  symmetry  prevents  any  explicit  comparison  with  the  solution  given 
in  this  paper  except  at  the  cost  of  labor  disproportionate  to  the  result. 

The  problem  must  have  an  extensive  domestic  history  in  the  schools:  Barbarin  charges  that 
E.  Catalan  was  among  those  who  have  proposed  it  as  an  elementary  exercise,  and  from  a  Russian 
scholar  the  writer  learns  that  it  has  been  there  extensively  used  in  the  schools  as  a  standard 
set-back  for  ambitious  young  geometers. 

■  See  Barbarin  in  Mathesis  (1896),  143-60;  also  BnU.  de.  S.M.F.  (1894),  76-80. 

'Pet.  Mem.,  XI  (1765). 

^L'lntermidiaire  (1894),  1-149;  also  in  Zeilschriftf.  M.  u.  N.  U.,  32.444. 


METHOD  AND  RESULTS 

The  eliminations  necessary  in  the  two  most  difficult  cases  are  accomplished  by  a  com- 
bination of  the  method  of  symmetric  functions  and  a  birational  transformation.  A  complete 
theory  of  the  role  of  birational  transformations  is  not  at  hand.  In  these  cases  one  of  the  two 
parameters  enters  the  resultant  in  the  first  order.  The  birational  transformation  reduces  a 
rather  complicated  equation  to  one  of  the  first  order  in  the  quantity  to  be  eliminated.  A 
mere  substitution  then  accomplishes  the  elimination.  In  a  more  general  case  it  might  always 
be  possible  to  reduce  by  such  a  transformation  to  the  order  in  which  the  parameter  entered 
(choosing  the  lowest  where  choice  is  offered)  and  materially  expedite  the  process.  Whether 
this  is  so  or  not  must  be  left  an  open  question.  In  this  case  such  a  surmise  led  to  a  tentative 
search  with  successful  results. 

The  determination  of  the  discriminant  at  least  as  to  its  algebraic  factors  was  conducted 
in  the  field  of  the  roots  (sides  of  the  triangles)  by  a  geometric  process  which  gave  two  factors 
afterward  transformed  by  the  transformations  of  the  elimination  to  the  field  of  the  param- 
eters.    It  then  became  possible  to  find  all  the  factors  and  check  the  result. 

The  separation  of  the  roots  was  effected  by  following  graphically  and  algebraically  the 
plane  of  the  sides  as  divided  by  discriminantal  lines  through  the  transformations  to  the  multiply 
covered  plane  of  the  parameters. 

In  the  internal  case  the  algebra  combines  with  the  case  of  three  internal  bisectors  (which 
has  a  unique  real  solution  for  real  data)  the  cases  of  one  internal  and  two  external  bisectors 
into  an  equation  of  order  ten  with  the  general  group.  No  further  irrationality  is  necessarily 
involved,  though  a  cubic  proved  a  convenience.  The  solution  is  explicitly  given  by  two  methods, 
one  with  and  one  without  this  cubic.  For  practical  approximation  a  tentative  method  not 
involving  an  elimination  proved  superior. 

In  the  external  case  the  solutions  are  not  always  real  for  real  data.  The  equation  is  of 
order  six  and  has  the  general  group. 

In  the  mixed  case  there  are  three  permutations  each  involving  an  irreducible  equation  of 
the  tenth  order  with  the  general  group.  The  separation  of  the  roots  is  accomplished  by  a 
simple  graph  and  approximate  solution  by  trial  with  the  table  of  logarithmic  sines. 

Special  cases  of  equal  bisectors,  isosceles  and  right  triangles  are  discussed.  In  each  case 
the  group  is  determined. 


I 

I.      THE    INTERNAL   PROBLEM 

To  construct  all  triangles  whose  internal  angle-bisectors  are  equal  to  three  given  lines. 

In  the  absence  of  a  geometrical  criterion  as  to  the  possibility  of  constructing  an  assigned 
figure  with  given  apparatus  (whether  ruler,  ruler  and  sect  carrier,  ruler  and  compass,  three- 
bar  link,  etc.)  resort  must  be  had  to  algebra. 

The  formulas  giving  the  internal  angle-bisectors  in  terms  of  the  sides  are: 

^'2=(«+*+c)  {-a-^b+c)bc 


M^ 


{b+cY 

)  Ka-b- 

•■+ay 

{a-\-b-\-c)  {a-\-b—c)ab 


T,     {a-\-b-\-c)  (a—b+c)ca  ,  . 

"-= ^^^ ^'^ 


(a+6)" 


It  is  to  be  noticed  at  the  outset  that  a  change  in  sign  of  a  leaves  K  unchanged  but  changes 
L  and  M  into  the  corresponding  external  bisectors  which  are  given  by: 

Tz.^     {a—b-\-c)  {a-\-b—c)bc 

K.      TZ T 

(h-cY 
j^     (a+b—c)  {—a+b+c)ca  ,  , 

^="  ic^^ ^'^ 

jrz^_  {—a-\-b-\-c)  {a—b+c)ab 
~  (a-b)' 

We  expect  then  an  algebraic  solution  of  the  problem  to  involve  as  necessarily  introduced 
extraneities  the  cases  where  the  assigned  quantities  appear  as  one  internal  and  two  external 
bisectors,  but  not  those  where  one  external  is  accompanied  by  two  internal. 

2.      THE    ORDER   OF   THE   PROBLEM 

Bezout's  rule  as  to  the  order  of  eliminants  gives  64  as  the  number  of  sets  of  values  of  a,  b, 
c,  satisfying  the  three  equations.  This  includes  as  improper  solutions  any  sets  of  values  which 
satisfy  them  independently  of  the  values  of  K,  L,  M.  That  the  number  of  such  improper 
solutions  should  properly  be  given  as  54  is  to  be  established. 

Geometrically  phrased  the  surfaces  denoted  by  the  equations  (i)  intersect  in  general  in 
10  distinct  points  which  vary  with  K,  L,  M,  and  have  at  certain  other  points  contact  of  various 
orders.  To  obtain  a  proper  enumeration  we  consider  slight  distortions  of  the  original  surfaces 
whose  intersections  will  be  in  fact  64  distinct  points.  Since  there  are  identical  relations  between 
the  planes  represented  by  the  separate  factors  such  a  distortion  must  destroy  these,  or  give 
independence  to  these  planes. 

Algebraically  phrased  we  write  for  a,  b,  c  and  the  other  plane  factors  quantities  differing 


2  THE   PROBLEM   OF   THE   ANGLE-BISECTORS 

arbitrarily  but  slightly  from  the  critical  values  and  determine  the  number  of  intersections  in  a 
region  near  the  critical  points. 

k  I  m 

and 

T=  7 — r-;— — ^,-— r-     and  take  a-\-b-\-c=i. 
{a+b+c)abc 

Then 

,     raii-aY      ,    Tb(i-by  Tc{i-cy 

I  — 2a  I  — 2b  1  —  2C 

and 

The  elimination  of  c  gives  three  quartics  so  that  the  64  points  are  still  represented. 
A  sufficient  independence  is  attained  if  we  treat  the  quantities  t,  a,  b,  c  as  independently 
variable  in  small  limits  near  the  values  determined  by  the  equations. 

The  point  a=i,  b  =  o,  c  =  o,  t=oo  is  critical.     Write  t=-,  a=i+a,  b  =  P,  c  =  y  and  con- 

sider  the  region  near  /  =  o,  a=o,  /8=o,  y=o. 
The  types  of  proper  approximations  are 

kt  =  a';  U  =  p;  mt  =  y. 

Hence  for  assigned  k,  I,  m,  there  are  two  intersections  in  the  region  and  this  point  is  to  be 
counted  a  double  point  for  every  k,  I,  m  since  the  values  of  a,  b,  c  are  in  the  limit  independent 
of  k,  I,  m. 

So  also 

a  =  o,  b=i,  c  =  o,  T=  CO 
and 

a  =  o,  b  =  o,  c=i,  T=oo. 

The  class  gives  then  six  intersections. 

The  point  a  =  i,  6=00,  c=  — <»,  t=o,  approached  by  b  =  p+^,  c=—p,  is  critical,  and 

writing  a  =  i+a,  b=^,  c  =  — ,  the  proper  approximations  are  of  the  types  ka=T,  l^=r, 
R         y 

m-f=T  with  four  solutions  and  six  members  of  the  class,  in  all  24  intersections. 

The  point  a=oo,  b=^,  c=— =0,  t  =  o,  approached  by  a=h  =  p,  c=i  —  2p,  gives  in  the 

same  way  proper  approximations  of  the  types 

with  8  intersections  in  the  region  and  three  members  of  the  class,  in  all  24  intersections. 

The  total  number  of  points  to  be  counted  for  the  contacts  at  these  critical  points  is  then  54. 

No  other  combinations  of  values  render  k,  I,  m  indeterminate  and  we  conclude  that  the 
order  of  the  problem  is  ten. 

3.      THE   ELIMINATION 

It  is  convenient  to  substitute  for  the  original  problem  that  of  constructing  a  triangle  whose 
internal  bisectors  are  in  a  given  ratio,  the  further  construction  being  an  affair  of  ruler  and  compass 
and  unique  save  as  to  cyclic  order. 


THE  ELIMINATION  3 

This  reduction  enables  us  to  take  any  relation  between  the  sides;  in  general  we  take 

a+b+c=  I. 

We  may  then  proceed  to  eliminate  c  by  means  of  this  and  b  by  Bezout's  or  other  methods, 
arriving  at  an  equation  for  a. 

The  consideration  of  such  an  equation  (which  is  obtained  in  §  15)  yields,  however,  no  view 
of  the  relation  between  sets  of  angle- bisectors  and  sets  of  sides;  to  obtain  which  the  symmetric 
function  transformation  of  order  three  is  introduced. 

a-\-b-{-c      =x=i 

'  ab-\-bc-\-ca  =  y  3) 

abc  =  z 

The  determination  of  any  quantity  of  which  y  z  are  rational  functions  gives  after  solution 
of  a  cubic  equation  a  unique  triangle;  that  is  as  to  ratios  of  the  sides.  It  turns  out  that  this 
cubic  though  convenient  is  not  algebraically  necessary,  the  character  of  the  irrationality  involved 
not  being  essentially  altered  by  its  introduction. 

Using  the  symmetric  function  n^ethod  we  write 

K^+L^+M'  =p 

K'L'+L'M'+I{'K'=q  (4) 

K' D  M'  =r 

and  expressing  p  q  r  in  terms  oi  x  y  z,  obtain 

g= (xy-zy^'^'^^~  syg+^'z-^^y].  (5) 


It  is  convenient  to  write 

'"k"      '~IJ'      "'~M 


*  =  ^.'      ^=fa'      '^  =  jr.-  (6) 


We  now  write  the  symmetric  functions  of  the  angle-bisectors  of  order  zero : 

q'  _  {k-\-l+mY 


«  =  -  = , 


pr    kl+lm+mk  '  (7) 

g^qi^{k+l+my 
'^     r'  Mm 


and  use  the  quantities  a,  j3  as  fundamental  parameters. 
Choosing  the  scale  by  placing  x=i 

(Ay-§yz+z-yy 


{yi- 2y'z+3yz- sz'-z)  Uy-8z-i)  '  (8) 

(4y'-8yz+z-yy 
{4y-Sz-iyiy-zyz' 


THE   PROBLEM   OF   THE   ANGLE-BISECTORS 


The  elimination  can  now  be  accomplished  by  means  of  a  birational  transformation: 

42  (y-  2Z) 


<t,= 


applying  which  we  have 


iy-z)  i4y-Sz~i)' 

z 
"(y-z)  Uy-&z-i)' 


4<^(.^+i)^(«^-t) 


(9) 


(lo) 


i6t4-(-t3(-4o<^+i6)+t"(33</>^-28<^)-|-t(-io<^3+io<^^)-|- </)"(</>+ 1)»' 
Substituting  for  t  and  writing  <^-|- 1  =  <t 

ibao-""— 4oaj8(r«-)-56a/8o-7-)-33a;SV'— 94a;8^o-s  (n) 

-|-(T4(6ia/3='-ioa;8'+4/83)  +  '^'(40"/8^-4i33) 

-|-a^(-50tt/33+a/8-l-4y8-l)-j-tr(2oa/33-  2a;84-f  S/S^) 

The  factors  a'  (cr—  i)  are  removed  in  the  course  of  the  work. 
This  equation  will  be  denoted  by 

.     F(o-,  a,  /3)  =  o 

and  referred  to  as  the  symmetrical  internal  equation. 

We  note  that  c  is  a  symmetric  function  of  a,  b,  c  and  that  if  o-  be  given,  4>,  t  are  uniquely 

determined,  v,  z  being  given  by 

-<A(</.-r+i) 

y= 


4t(<^— T—  i) 

4t(<^-t— i) 

which  is  unique  except  for  the  singular  points  and  lines  of  the  transformation: 

The  lines  y—z  =  o(  ,.       ^    ^,  .   ^   , 

/  (  correspondmg  to  the  pqmt  </>=  =o  t=  oo 

4y— 52 —  I  =  o  1 

the  line  y— 22  =  0  corresponding  to  the  point  <^  =  o     t=  — i 

the  line  z  =  o  corresponding  to  the  point  <^  =  o     t  =  o 

and  inversely 

the  line  <^  =  o  corresponding  to  the  point  y  =  o    2  =  0 

the  line  t  =  o  corresponding  to  the  point  y=  1=0   z=  00 

the  line  <^— t— 1  =  0  corresponding  to  the  point  y=      22=  =0. 

If  instead  of  choosing  x=  i  as  defining  the  scale  we  take 

4xy— 8z— a:J=i  and  write 
xy—2Z=K    whence  x'  =  4K— I 

we  have  by  elimination  of  x  and  y 

[2(4-c+i)  +  k]» 


^= 


2^(8k-i)  +  2(s-c-i)  +  k3' 

[2(4K+i)+k]3 
(4/C-1)   (2+t)'z' 


(9O 


(12) 


THE   PARAMETRIC  FIELDS 


A  simpler  birational  transformation  is  now  available 


with  reversions 


4(cz  (4*^-  i)z  ,s 

^  tf, 


The  expressions  for  a  ft  and  the  eliminant  are  the  same  as  before,  the  same  extraneities 
occurring. 

This  transformation  is  simpler  but  has  a  practical  inconvenience  in  the  determination  of 
X  from  the  cubic  x^  =  4k—i. 

The  triangles  for  which  x  —  Xi,  wXi,  oy'x,,  are  not  distinct  in  ratio  of  sides. 

We  shall  use  this  transformation  to  obtain  certain  information  but  shall  not  discuss  it 
completely. 

The  extraneous  factors  <^(<^+i)^  need  notice. 

<^  =  o  leads  to  a' =i  7=02  =  0 
a  '■  b  '■  c=i  :  o  :  o 
K  ■  L  '■  M=  1:0:0 

which  is  not  in  general  a  solution. 

<^=  —  I  leads  to 

o  :  6  :  c=  I  :  «> :  <»',  («>'=  i)  a  complex  triangle  whose  angle-bisectors  (internal)  all  vanish  and 
which  though  a  solution  of  the  ratio  problem  by  way  of  indeterminateness  is  not  a  solution  of 
the  original  problem. 

4.      THE   PARAMETRIC    FIELDS 

Proceeding  to  the  discussion  of  the  equation 

we  note  that  it  is  a  two-parameter  equation  of  order  10  irreducible  in  the  domain  of  rationality 
constituted  by  a  ^  and  their  rational  functions.  For  u  occurs  only  to  the  order  i  and  any 
reduction  in  the  domain  must  be  of  the  form 

F=MiPa+Q) 

where  M,  P,  Q  are  rational  functions  of  /3,  a-  alone.  Hence  the  terms  containing  a  and  those 
free  from  a  have  a  common  factor — which  is  not  the  case. 

The  problem  can  be  discussed  in  three  fields: 

i)  a,  /?  may  be  restricted  to  such  values  as  arise  from  the  assignment  of  real  quantities 
as  the  lengths  of  the  angle-bisectors.    This  may  be  called  the  field  of  real  angle-bisectors; 

2)  a,  /3  may  be  any  real  quantities; 

3)  1,  y3  may  be  any  algebraic  quantities. 

In  connection  with  the  first  field  we  have  a  part  of,  and  in  connection  with  the  second 
field  the  whole  of,  a  real  surface  which  in  virtue  of  the  fact  that  a  is  single-valued  may  be  con- 
sidered as  a  geographical  surface  and  is  properly  represented  by  a  set  of  a  contour  curves  cover- 
ing the  whole  y3,  <r  plane. 


6  THE  PROBLEM   OF   THE   ANGLE-BISECTORS 

In  connection  with  the  third  field  we  have  a  "  variete"  in  four  dimensions  to  use  the  phrase 
of  MM.  Picard  and  Simart  and  in  accordance  with  their  theory  since  the  equation  is  the  result 
of  ehmination  and  of  numerical  genus  zero  and  geometric  genus  zero,  can  be  birationally  trans- 
formed into  the  totality  of  point  pairs  of  two  Neumann  spheres.  This  is  accomplished  in  this 
case  by  means  of  the  parameters  <f>,  t  in  terms  of  which  o-,  a,  /3  are  rationally  expressed  (lo)  and 
which  are  themselves  rational  in  o-,  a,  j8,  viz.: 

<^  =  <r-i  ,  ■^=Q  ■ 

In  the  field  of  real  angle-bisectors,  the  quantities  k,  I,  m  are  necessarily  positive  and  o  /8  are  posi- 
tive and  restricted  to  a  certain  region  of  the  real  a  /S  plane. 


ds) 


In  fact,  k,  I,  m  are  the  roots  of  the  equation 

ru'—qu^+pu—i  =  o  , 
and  p,  q,  r  are  positive. 

The  discriminant  expressed  in  terms  of  a,  fi  is 

/?■(»— 4)  + 20^(9— 2a)/3— 27a3 

__  ^ 

which  must  be  positive  for  real  positive  roots. 

The  quartic  curve  along  which  the  numerator  vanishes  will  be  denoted  shortly  by 

Z)(a,;8)  =  0. 

It  has  a  cusp  of  the  first  species  at  (o,  o)  where  the  tangent  is  /?=  o,  a  cusp  of  the  first  species  at 
(3,  27)  where  the  tangent  is  parallel  to  2'ja=/3;  asymptotes  0=4  and  /8=  —  ^;  a  parabolic 
asymptote  /8=4a';  and  a  real  finite  inflexion  without  apparent  significance  in  the  problem. 


i 


TWO   SPECIAL   CASES  7 

In  making  a  diagram  for  psychologic  reasons  connected  with  the  single  valued  property 
of  a,  we  reverse  the  usual  alphabetic  order  and  take  a  vertical.  The  region  inside  the  cusp  in 
the  first  quadrant  is  the  field  of  real  angle-bisectors.  Inside  the  hyperbolic  branch  in  the  sec- 
ond quadrant  and  between  the  /?  axis  and  the  curve  in  the  fourth  the  discriminant  is  also  posi- 
tive, but  as  either  a  or  ^  is  negative  the  regions  represent  real  squares  of  angle-bisectors,  some 
of  the  set  being  negative.  The  curve  D{a.,  /3)  =  o  in  a  proper  sense  and  the  axes  in  an  improper 
sense  are  loci  of  equal  angle-bisectors  (Fig.  i). 

S.      TWO   SPECIAL   CASES 

At  this  stage  it  is  convenient  to  solve  some  special  cases. 
First  the  case  of  equal  bisectors ;  K=L  =  M    a  =  3     13— ly. 
On  writing  <r=$s,  F(cr,  a,  j8)  =  o  becomes 

165'°— i2o5'-|-5657-|-2975'— 28255— 1735^+34853— 1775^+385— 3  =  0, 

which  reduces  to 

(25-3)  (5-1)3  (25^+35-1)3  =  0. 

5=  I  gives  the  equilateral  triangle. 

5=  I  gives  K=^  s=  00  (using  the  k,  z  chain)  the  sides  being  a  :  6  :  c ::  i  :  00  :  00  ;  on 
the  scale  a-\-b+c=  1     a  ■  b  ■  c-.^  ■  m  :  —m-\-\  where  m  is  an  infinite  quantity. 

This  is  the  limit  of  a  triangle  with  very  small  base  and  very  long  sides  when  the  vertex  is 
brought  into  line  with  the  base. 

Since  c  is  negative  the  B  and  A  bisectors  are  external. 

This  triangle  is  a  constantly  occurring  triviality. 

—  S— 1-^17    ■  T  "i+l'  17     5+1-^1^7 

5=—^^ — ' — -'  gives  a:b.c::—i:  ^-^-^ — -  ■  ^-'-^ — - 

4  4  4. 

an  isosceles  triangle  with  two  external  bisectors  equal  to  one  internal.    The  angle  A  is  approxi- 
mately 25°  20'. 

5=     ■^     ' — -'  gives  a  ■■  b  ■■  c::—i  ■■  - — - — -  :  ^ — - — ' 
4  4  4 

which  are  impossible  in  the  sense  that   |6|+|c|<  la]. 

As  the  class  will  need  frequent  discussion  we  reserve  the  word  "impossible"  for  this  exact 
meaning. 

The  case  a  =  4  ^=  54;  the  point  in  the  (a,  /3)  plane  being  the  intersection  of  the  asymptote 
of  Z)(a,  )8)  =  o  with  the  curve. 

K  :  L:  M::2  :  2  :  i 

(7=0  is  a  root  by  inspection.     Writing  (t  =  35  as  before  the  remaining  terms  reduce  to 

(253+25^-75+2)^  (^—2)  (5^— 5)  =  o. 
5=2gives<|)=5,  T=4,  y=2z=oo. 

This  is  the  same  infinite  triangle  as  in  the  case  a=3  /3=27,  a  different  path  of  approach  to  the 
limit  being  involved  in  the  different  ratio  of  bisectors. 
Using  the  magnitude  of  o-  as  an  index  this  root  is  (2). 


THE   PROBLEM   OF   THE   ANGLE-BISECTORS 


^  =  0      <^=    — I       T=0      y=<X>       2=00       -=o 

2 


This  is  the  limit  oi  a  -.h  :  c::m  :  m  :  — 2m+i  when  m=  oo  . 
The  root  is  (7). 

•5=1/5   gives  a  -.h  :  c::2  :  \   \—\  :  v'S— i  the  obtuse-angled  triangle  which  occurs  in 
Euclid's  construction  of  a  regular  decagon.      The  root  is  (i). 

^=  — 15     a  :  b  :  c::  —  2  :  j   s-|-i  :  y  s+i  the  acute-angled  triangle  of  the  same  context. 
The  root  is  (8). 

The  cubic  factor  is  irreducible  and  has  three  real  roots  which  are  to  be  counted  twice  each. 
Approximations  are 

5=1.  2098        a  :  6  :  c : :  1 .  947  :  .  584 
5=    .3259  1.247  :    045 

^=  -2-5357  665  :  .559 

The  triangle  corresponding  to  (5)  (6)  is  impossible. 


-I -531  (3)  (4) 

-  •  202  (S)  (6) 

-  233  (9)  (10) 


6.      REALITY    or    THE    SOLUTIONS 

Returning  to  the  general  equation  F{<t,  a,  y8)  =  o  it  is  first  to  be  remarked  that  as  o-  is  a 
symmetric  function  of  a,  b,  c  only  real  values  of  o-  can  lead  to  real  triangles;  also  that  real  values 
of  0-,  provided  that  a  (3  are  within  the  field  of  real  angle-bisectors,  always  lead  to  real  triangles. 
For  real  o-  and  real  fi  involve  real  <^  and  t  and  so  real  y  and  2.  Hence,  a,  b,  c,  are  either  real  or 
one  is  real  and  the  other  two  are  complex  conjugates.  But  if  a  =  b,  K  =  L  and  D(a,  fi)  vanishes. 
Now  the  region  of  the  a,  /3  plane  where  K,  L,  M  are  real  is  finitely  separated  from  the  regions 
where  the  squares  of  these  quantities  are  real  and  in  part  negative,  the  intervening  regions 
having  some  of  the  squares  complex.  It  must  be  inferred  then  that  real  sides  and  real  bisectors 
are  only  found  together  in  the  single  region  inside  the  cusp  of  D{a  13)  =  0,  this  curve  being  dis- 
criminantal  for  both  sets  of  quantities ;  and  that  a  real  o-  inside  this  region  means  a  real  triangle. 

It  is  proper  to  note  however  that  this  argument  leaves  open  the  possibilities  that  (i)  other 
triangles  besides  the  isosceles  class  occur  on  the  locus  D  =  o;  (2)  complex  or's  and  complex  tri- 
angles may  occur  inside  the  cusp;  (3)  real  sides  may  occur  outside  this  region,  namely  with 
negative  squares  of  bisectors;   (4)  real  triangles  inside  the  cusp  may  be  impossible  triangles. 

Of  these  propositions  (i)  (3)  (4)  are  affirmed  and  (2)  is  denied  by  the  subsequent  develop- 
ments. 

7.      THE   ELEMENTARY   THEORY   OF   EQUATIONS   APPLIED 

The  next  undertaking  is  to  use  as  far  as  possible  the  elementary  theory  of  equations  in 
separating  the  roots,  determining  their  reality,  and  classifying  the  triangles  connected  with 
them. 

On  account  of  the  labor  involved  in  some  of  the  operations  (such  as  determining  the  set 
of  Sturm's  functions)  being  prohibitive,  a  complete  discussion  cannot  be  had  by  this  method, 
but  the  results  as  far  as  they  go  are  valuable  in  themselves  and  as  a  corroboration  of  the  sub- 
sequent treatment  of  the  problem. 

We  consider  first  the  functions 

,,,  =  o-3-j8o-K  (8  (16) 

j7j  =  o-3— y8o--|-2j8 


/ 


J 


ELEMENTARY   THEORY   OF  EQUATIONS  APPLIED 


Using  the  k,  z  chain 


x^= 


a 


z  = 


4Vt 


(17) 


Assuming  a  y3  to  lie  within  the  real  angle-bisector  field 

a^i      ^>27;    D(a,l3)>0 

a  change  of  sign  of  17,,  rj^  involves  a  change  of  sign  of  x  and  z,  and  so  by  means  of  17,,  rj^  the 
roots  are  classified  with  respect  to  their  connected  triangles. 

17,  =  0  and  >;2  =  o  have  each  three  real  roots,  one  negative,  for  18—27. 

Considered  as  curves  in  the  a-,  -q  plane  the  families  for  varying  P  have  common  properties 
and  one  case  may  be  taken  as  a  type.  The  graph  shows  1?,,  172  for  /?=  54  and  also  F{<r,  4,  54) 
(Fig.  2). 

F 


By  reducing  F(a-a  ^)  to  the  second  order  by  means  of »;,  =  o  and  172  =  o  respectively  it  appears 
that  for  the  roots  of  the  77's,  for  <r  =  o,  and  <t=  i,  provided  that  a^^  /8— 27,  F  has  the  following 
signs : 

A  B  o  cr=i  C  D  E  F 

+  +  ±  +  +  +  -  + 

The  sign  at  o  is  +  if  a  >4  and  —  if  a<  4. 

F=o  has  then  at  least  one  real  positive  root  between  E  and  F. 

Since  here  x,z,  and  hence  y  i=[x'+&z+i]4x),  are  positive  the  problem  has  a  real  solution 
to  be  interpreted  as  referring  to  three  internal  angle-bisectors. 

The  signs  of  x,z  in  the  intervals  —  =0  to  yl,  .4  to  5,  etc.,  are 

A  B  o  <T=i  C  D  E  F 


X 

z 


+ 


+ 


+ 


+ 
+ 


+ 


+ 
+ 


+ 


The  intervals  CD  and  EF  are  the  only  ones  in  which  such  an  internal  solution  can  occur. 
No  conclusion  can  be  drawn  as  to  CD  for  Fia-)  is  positive  at  both  C  and  D. 

The  greater  root  of  V2  is  a  superior  limit  for  roots  of  F.  A  superior  limit  not  so  close  but 
more  convenient  is  furnished  by  ]/  /3. 


lO 


THE   PROBLEM   OF  THE   ANGLE-BISECTORS 


We  next  proceed  to  transform  the  equation  so  as  to  find  the  number  of  real  roots  for  inter- 
vals in  which 

(i)  iji  r?2  are  both  positive, 

(2)  i?i  1/2  are  negative  and  positive  respectively, 

(3)  Vi  '72  are  both  negative. 

For  arithmetic  convenience  we  begin  by  eliminating  o-  from  F{(t  a  /3)  =  o  and  o-3  =  /8(<t— X). 
F  is  first  reduced  to  a  quadratic;  that  is  a  function  M  rational  and  integral  in  o-,  a,  /8  is  deter- 
mined such  that 

F{<J  a  P)-M[cTi-P{a-X)\  =  N 

where  N  is  of  the  second  order  in  o-. 

This  is  conveniently  done  in  a  tentative  fashion  and  no  extraneities  are  introduced,  for 
the  equivalent  end  term  process  is 

I  F{<r  a  13)  o 

-M         o-3-/3(o— A)         I 

and  has  the  determinant  of  multipliers  i. 

Thence  the  result  is  obtained  by  substituting  in  the  general  eliminant  of  a  cubic  and  quad- 
ratic form.    To  use  the  end  term  process  again  would  introduce  an  extraneous  factor. 

The  arithmetic  is  simplified  by  using  ir  =  A— i  instead  of  \. 

The  result  arranged  as  a  polynomial  in  v  is 


ir~ 

TT* 

ir< 

irT 

n* 

ITS 

jr* 

1,1 

IT" 

IT 

z 

a  (33.  . 

....     1 

-16 

0 

0 

0 

0 

P'.. 

64 

0 

0 

0 

0 

a3j3».  . 

81 

-36 

22 

-4 

I 

0 

0 

a'p'.  . 

-480 

260 

-76 

12 

-4 

0 

0 

affi.. 

768 

-512 

-256 

-16 

0 

0 

0 

Ifi.. 

64 

0 

0 

0 

03/S  .. 

-1152  . 

800 

968 

-354 

228 

0 

4 

2 

0 

a^/S  .. 

3072 

-2560 

-2816 

1760 

388 

—  140 

12 

-4 

0 

oA     .  . 

i 

1.096   —  2( 

h8 

-5376 

3328 

1648 

—  1224 

-147 

168 

-6 

-8 

I 

(18^ 


For  the  investigation  of  region  (2)  where  the  values  of  A  range  from  i  to  2  we  write 

_i  — TT     2  — A 
''^"T^^A^  • 

The  equation  in  p  will  then  have  positive  roots  for  roots  of  F(<r)  in  the  region  (2)  that  is 
between  the  curves  r),=o  and  172  =  0. 
The  equation  in  p  is: 


p.. 

P' 

pf 

p' 

p' 

p' 

f 

f 

p" 

P 

I 

a/33... 

-16 

—  96 

—  240 

-320 

—  240 

-96 

—16 

/33 

64 

384 

960 

1280 

960 

384 

64 

a3ft' 

I 

4 

22 

68 

161 

320 

400 

256 

64 

a'/P 

-4 

—  20 

—  104 

-168 

-180 

-644 

-1280 

—  1024 

—  288 

alP 

-16 

-368 

-2384 

—  6192 

-7728 

-4688 

-1136 

—16 

ti' 

64 

448 

1344 

2240 

2240 

1344 

448 

64 

a>ti 

2 

22 

104 

508 

1490 

3150 

6084 

7120 

3616 

496 

a'fi 

-4 

-24 

-188 

-592 

1316 

7240 

6972 

-832 

-2592 

—288 

ai 

• 

I 

2 

-33 

-48 

399 

330 

—  2015 

-268 

3168 

—  2160 

432 

(19) 


ELEMENTARY  THEORY  OF   EQUATIONS  APPLIED  II 

The  transformation  is  of  course  effected  in  the  coeflScients  of  the  powers  and  products  of 
a,  /3,  separately. 

Proceeding  with  the  determination  of  signs  of  the  coefficients: 

[p'"]  a3  is  positive  in  the  region, 

[/o']  2a»(a+o/3—  2/3)  is  positive, 

[p']  and  [p'l  vary  and  are  reserved  for  later  discussion. 

[p']    This  coefficient  can  be  arranged  as 

-i6(|8+i)I>+a3(-/8»+i2y8-33)+/8^(-4ia3+i84a'-3S2«+384)-304«'^ 

where  D  is  written  for  D{a  /3)  which  is  positive  in  the  region. 

Since  a^2>  and  P^2j  this  is  negative  throughout  the  region. 
[p5]  can  be  arranged  as 

-ga/SD-zS'CsiSaJ- 15600^+23840- i344)-/3(iio2a3-i3i6a^)  +  a3(-/3»+ 330) 

which  is  negative. 

Mis 

—  240/8Z)— ^(79903— 41400'+ 6192a— 2240)  — y8(333oa3— 72400^)  — 201503 

which  is  negative. 

[p3]  is 

—  320i8Z)— 18^(96003- 5ii6a»+7728o— 2240)  — 18(255603— 6972a')— 26803 

which  is  negative. 

[p']  is 

-24o()8+i)D-i28(a-3);8'- (4320)8-34880)0^-32003/3-331203 

which  is  negative. 
[p]is 

(-96/3+8o)£»+64(a-3)/3[/3(-o'+Ioa-32)  +  a'(-j8+2l)] 

which  is  negative. 

[p»]  is 

-i6(/8+i)£> 
which  is  negative. 

Returning  to  the  eighth  and  seventh  powers, 

[p7]-4[p8]  =  4[-4Z)+4o'/3(-a+3)+4o'y8'(-^+io)-87a3] 

which  is  negative. 

Hence  if  [p*]  is  negative  [p']  is  also. 

Of  the  eleven  signs  either  the  first  two,  three,  or  four  are  positive  and  all  others  negative 
throughout  the  region  and  F{p)  =  o  has  one  and  only  one  positive  root. 

From  the  last  result  it  follows  that  F{cr,  o,  /3)  =  o  has  one  root  and  only  one  root  in  the 
intervals  A  •  •  •  B,C  •  •  •  D,  E  •  •  •  F  and  since  there  is  one  root  in  £  •  •  F  which  leads  to  a  tri- 
angle with  positive  sides  and  therefore  internal  bisectors  in  the  given  ratios  and  since  all  such  roots 
fall  in  these  intervals  there  is  one  and  only  one  solution  of  the  original  geometrical  problem  for  every 
set  of  real  quantities  assigned  as  the  lengths  of  the  internal  angle-bisectors.  Since  this  root  is  greater 
than  3  and  less  than  |  ''/S  or  less  than  the  greatest  root  of  0-3- j8<r+2/3=o  and  no  other  root 
occurs  in  the  interval  there  can  be  no  trouble  in  separating  the  root.  The  sides  are  to  be 
determined  from  <t  by  rational  operations  and  solution  of  the  cubic  (P— i'+^y— 2  =  0). 


12  THE   PROBLEM   OF   THE   ANGLE-BISECTORS 

In  a  certain  narrow  sense  this  is  a  solution  of  the  problem,  but  it  is  necessary  to  inquire 
more  closely  as  to  the  character  of  the  irrationality  involved  and  as  to  the  necessarily  intro- 
duced extraneities  and  their  connection. 

Proceeding  with  the  method  in  hand  we  write  >l/=k—  2. 

The  equation  in  1/'  will  have  positive  roots  corresponding  to  roots  of  F=o  in  the  intervals 

—  00  to  j4  and  D  to  E.    As  however  the  least  root  of  772  =  o  is  an  inferior  limit  the  interval 
D  •  -  Eis  the  only  one  in  question. 

In  this  case  the  labor  required  is  not  justified  by  the  results  to  be  expected,  and  it  may 
merely  be  noted  that  the  coefficients  of  i/*'"  and  i/''  are  positive  while  the  constant  term 
is -i6(,8+i)ZJ. 

From  which  at  least  one  root  results. 

By  writing  0= —  we  have  positive  ^'s  associated  with  roots  of  F=o  in  the  intervals 

TT 

B  •  •  C  and  F  •  •  •  -\-<x  but  as  the  greatest  root  of  1;,  =  o  is  an  upper  limit  the  interval  B  •  •  C  is 
alone  possible. 

Here  taking  the  coefficients  from  the  polynomial  in  tt  with  the  proper  changes  in  order 
and  sign  •  — 

[6^0]  is  ai  and  is  positive, 

[0*]  is  8a3-|-4a^;8—  2a3;8  and  is  always  negative, 

This  varies  in  sign,  being  negative  at  (3,  27)  and  positive  at  (4,  54). 
[6'!]  is  4a3fi'-^a'p^+4D+{i6a-^y)a^/3+66ai]  and  is  positive. 

[^6]  is  -  l6|8D-f-,5^(-42a3-f2I2a'- 2S6a)-|-a»;8(- 294a-f  388)- I47a3. 

The  coefficient  of  /8"  has  the  value  6  for  0=3  but  rapidly  diminishes,  the  derivative  being 

—  118,  and  has  negative  values  for  «— 3.05  .  .  .  ,  while  /3  is  very  near  27  if  this  coefficient  is 
positive.     Hence  the  whole  expression  is  easily  seen  to  be  negative. 

[0^]  varies  in  the  region. 

[6^]  is  (8ia^-48oa-|-768)a|8^-(- (968a- 28i6)a»j8+ 164803  and  positive. 

[^3]  is  32a»(- 2Sa;8+8o/8- 104a). 

The  coefficient  of  /8  is  negative  for  a  >3 . 2  when  /8<  40. 
Hence  the  whole  expression  is  less  than  —  1 1 2  (3  2a') . 

[6^]  is  384a2(— 3a/3-f  8j8—  14a)  and  negative. 

The  remaining  coefficients  are  positive. 
The  signs  then  run  in  the  interval 

+   -±   +   -±   + +  + 

The  variations  of  [0^]  and  [0^]  have  then  no  effect  and  there  may  be  always  six  roots  in  the 
interval. 

With  these  results  the  practical  utility  of  the  classical  theory  seems  to  end.  The  direct 
evaluation  of  the  discriminant  in  Bezout's  form  is  impracticable,  the  elements  of  the  ninth  order 
determinant  involving  a  and  /3  in  polynomials  of  the  fourth  order  with  coefficients  of  six  figures. 
Still  less  ie  it  to  be  expected  that  a  successful  attack  could  be  made  on  Sturm's  functions. 


REALITY   OF   THE   ROOTS   FOR   REAL  ANGLE-BISECTORS 


13 


Recourse  must  then  be  had  to  special  methods  applicable  to  this  problem  and  to  the  power- 
ful general  method  of  following  the  transformations  used  in  the  elimination  and  interpreting 
them. 


8.      REALITY   OF    THE   ROOTS   FOR   REAL  ANGLE-BISECTORS 

If  the  bisectors  are  real  k,  I,  m  are  positive.    Write  t={a-\-b-{-c)abc  and  3'i  =  y,,  y'  —  yt 

yj  =  — ,  the  internal  formulas  become 
tnt 


1  —  2* 


y=— 7 r- with  a;=a  for  Ti;  x=b  ior  y^:  x=cioTyi 

x{i  —  x)' 


(20) 


The  curve  represented  by  this  equation  has  the  line  x=i  for  an  asymptote  with  a  cusp  at 
infinity.  The  x  axis  is  also  an  asymptote  with  an  inflexion  at  infinity,  the  curve  approaching 
with  negative  y.  The  y  axis  is  an  ordinary  asymptote.  The  curve  meets  the  x  axis  also  at 
«=HFig-3)- 


0 

\                                               ^ 

^^^c. 

W     :                 CX'^ 

\ 

1b.  ;         /b. 

A>\ 

Iax   ':            ^3 

"^ 

'■■                       Fig  3 

Consider  first  t  positive  and  therefore  y  positive. 

For  any  I  the  lines  ^i  =  t".  ;  y'  —  Tt'  >'3  ~  ~i  ^^^^  '-^^  ^^^  curve  in  one  real  point.    The  sum 

rSt  It  fHt 

of  the  abscissas  is  |  when  t=  °o  and  is  o  when  t  =  o.  Since  the  curve  monotonously  approaches 
the  y  axis  as  t  decreases  the  value  i  is  reached  once  and  only  once  by  a-\-b-\-c;  hence  there 
is  always  one  and  only  one  solution  for  a  positive  /. 

The  original  problem  restricted  to  internal  bisectors  has  a  unique  solution,  for  it  is  evi- 
dent that  a  b  c  and  t  must  all  be  positive  for  this  interpretation.  As  to  the  possibility  of  the 
values  a  6  c  we  note  that  since  the  perimeter  is  i  and  each  side  is  less  than  J  any  two  must  be 
together  greater  than  the  third. 


14  THE   PROBLEM   OF   THE   ANGLE-BISECTORS 

Considering  negative  t's  each  of  the  lines  yi  =  T,!  ^tc,  cuts  the  curve  in  three  real  points. 
Assuming  that  k<l<m  we  call  the  abscissas 

ai<a2<a^;    6i<62<6,;    c,<C2<Cj. 

The  product  of  27  factors 

P[i-iai+bj+Ck)]    i,j,k=i,2,s 

is  symmetric  in  the  a's,  b's,  and  c's  separately  and  can  be  expressed  in  terms  of  k,  I,  m,  t. 

This  product  vanishes  for  solutions  and  only  for  solutions. 

From  the  other  eliminations  we  know  that  there  are  in  general  ten  finite  values  of  /  satis- 
fying P=o.     We  have  seen  that  one  and  only  one  is  positive. 

For  negative  t's  we  consider  the  27  combinations  ai-\-bj-\-Ck  denoting  them  shortly  by 
{i,j,  k)  understanding  that  the  indices  in  the  order  written  refer  to  a,b,  c. 

Some  of  the  27  are  excluded  as  being  continuously  greater  or  less  than  i.    These  are 

(i,  I,  i)<o 

(2,  2,.2)>f 

(2,   2,3),    (2,3,   2),    (3,   2,  2)>2 

(2,3,3)>  (3,  2)3),  {2„2>,  2)>f 

(3,  3,3)>3 

(i,  I,  2),  (i,  2,  i),  (2,  I,  i)<i 

There  remain  15  arrangements  which  may  possibly  yield  solutions. 
Taking  the  set  (i,  2,  2)  aj-t-6j-|-C2=2  for  t  =  o,  y=  =0 

=  —  00  for /=  00,  3^  =  0. 

Hence  at  least  one  real  root,  for  the  abscissas  all  decrease  monotonously  with  increasing  /, 
and  are  finite  and  continuous  in  the  interval  o>/>  —  00 . 

Similarly  (2,  i,  2)  and  (2,  2,  i)  each  yield  at  least  one  real  root. 
Of  the  six  permutations  of  (123)  three  can  be  excluded,  namely: 


(i,  2,  3) 
(i,  3,  2) 
(2,  1,3) 


(a,  -l-62-f  c,)  >  (c, -I-C2-I-C3)  >  2 
{a,^b,+c,)>{]},^-b,+b,)  +  {c,-b,)>2-\>i 
(a2+6i+C3)>(c,-|-C2-fc3)>2. 


For  (2,  3,  i)  a2-\-bi-\-Ci  =  2  for  l  =  o;  for  /=—<»,  y=  — o,-the  equation 

X^  —  20<^-\-x{l-]r2kt)  —  kt  =  O 


has  one  root  J  and  the  others  approach  infinity  with  ±  1    —  2kt.  

As  k<l<m,  a2+bi-\-Ci  approaches  negative  infinity  with  i+y'  —2U  —  V  —2int. 

Hence  (231)  yields  at  least  one  real  root. 

In  a  similar  manner  it  can  be  shown  that  {312)  and  (321)  each  give  at  least  one  real  root. 

In  the  set  of  permutations  of  (113)  we  have  in  the  case  (131)  ai-\-b^-\-Cx  =  i  for  /  =  o  and  is 
negatively  infinite  when  t  is. 


REALITY  OF   THE   ROOTS   FOR  REAL  ANGLE-BISECTORS  1 5 

To  determine  whether  it  ever  exceeds  i  in  the  range  we  express  the  roots  of 

3;^  — 2.r'-|-(i-|-2^/).T— ^^  =  0  (21) 

in  terms  of  kt  near  x  =  o,t~o  and  x=i,  t  =  o. 

x,=ki-k^t?-2kH*  ....  (22) 

^2  =  1— 1    —kt-\ h  .  .  . 

2 

a;,  =  1+1    —kl-\ h  .  .  . 

2 

a,-\-bj-\-Ci=kt-\-i-\-\    —U+mt+  .  .  . 

Since  in  the  limit  as  /  approaches  —  o  the  square  root  term  exceeds  in  absolute  value  the 
sum  of  the  terms  of  the  first  order,  (131)  is  greater  than  one  in  the  region  of  /=  — o,  and  so  gives 
at  least  one  real  root.     Similarly  (311)  gives  a  real  root  at  least. 

The  treatment  is  not  sufficient  for  (113)  as  this  may  not  approach  minus  infinity  with  /. 
Reserving  this  case  and  taking  the  i,  3,  3  set 

(133)  :  a,+b,+Ci>c,+b3+c^>c,+C:,+Ci>2 
(313)  :  ai+b,  +  c,>b,+b,+b3>2 

(331)  requires  a  more  detailed  examination  and  is  to  be  considered  in  connection 
with  (113). 

(113)  a,+&,+C3  =  I  when /  =  o 
(331)  03+^3+^1  =  2  when /  =  o. 

When  t  approaches  minus  infinity  (113)  approaches  ^  —2t[  —  ]/k  —  }^  1+]  m]  while  (331) 
approaches  the  same  quantity  with  reversed  sign. 

So  one  and  only  one  of  the  two  combinations  gives  at  least  one  real  root  in  the  interval. 

In  terms  of  the  fundamental  parameters  1  ^+1  l>ym  reads  a<4  and  in  this  case  (113) 
has  the  root,  but  if  a  >  4  (33 1 )  has  it. 

The  boundary  case  0  =  4  needs  special  mention  as  in  this  event  the  first  approximation 
is  indeterminate. 

Consider  the  limit         (  — 1/^  — 1   /+]/w)(l    — 2/) 

t=  —  00 

where  k^  is  such  a  number  that  \/ki-\-\  'l—\/m  =  o. 
We  have 

Limit  =  limit        {-y'kV+x-Vl+\/M)y2 

x=o  yy 

y  =  o 
-J^  limit  ^. 


If  then  the  path  of  approach  be  along  the  parabola  2x^  =  k,y  the  limit  is  i.     So  we  may  consider 
that  (113)  has  a  root  at  the  limit.     Or  by  approaching  on  the  other  side  (331)  has  a  root  here. 


l6  THE  PROBLEM   OF   THE   ANGLE-BISECTORS 

For  a  =  4  the  original  equation  has  a  root  cr  =  o  which  leads  by  the  chain  of  equations  to 
just  such  an  infinite  triangle  as  is  here  in  question. 

In  every  case  then  there  are  at  least  ten  real  roots  for  finite  values  of  t,  and  as  we  know  that 
there  are  only  ten  such  roots  there  must  be  one  in  each  of  the  classes  specified. 

The  arrangement  of  the  bisectors  in  the  triangle  should  be  noted.  For  the  internal  case 
the  magnitudes  of  the  bisectors  and  the  sides  which  they  meet  are  in  reversed  order. 

For  the  case  (231)  the  internal  bisector  is  M  since  c^  is  negative. 

The  least  bisector  is  internal  but  no  further  statement  can  be  made  as  to  absolute  magni- 
tudes, for  if  /  is  large  c,  may  in  absolute  value  exceed  6,. 

For  (312)  the  medium  bisector  is  internal,  for  (321)  the  least.  So  (122)  has  the  greatest 
internal,  (212)  the  medium,  and  (221)  the  least. 

The  set  i,  i,  3  are  all  impossible  triangles  (see  §  11)  and  the  words  internal,  external  are 
without  a  proper  meaning,  but  we  may  say,  in  (131)  the  medium  bisector  is  associated  with  the 
side  that  is  unique  in  sign  {b). 

In  (311)  the  greatest  and  in  (113)  the  least  is  so  associated.  If  a>4  (331)  gives  a  possible 
triangle  with  the  least  bisector  internal.  The  want  of  symmetry  of  the  arrangement  by  which 
the  least  bisector  is  "internal"  in  five  analytic  cases  out  of  nine  and  in  four  real  cases  out  of 
eight  (or  in  three  out  of  seven)  is  to  be  noticed. 


9.      A   PRACTICAL   METHOD   FOR   APPROXIMATE    SOLUTIONS 

Following  the  plan  of  the  last  section  we  have  the  three  equations  (21)  with  k,  I,  m  given 
and  a  particular  set  of  three  roots,  say  a^,  b^,  d  whose  sum  is  to  be  unity.  For  the  sets  (3,  2,  i) 
(2,  3,  i)  (3, 1,  2)  (i,  2,  2)  (2, 1,  2)  (2,  2,  i)  it  is  easy  to  show  that  the  sum  decreases  monoto- 
nously as  t  increases.  For  the  other  negative  sets  this  is  not  so,  but  the  solution  occurs  on  such 
a  slope  which  is  unique.     By  trial  we  may  find  values  of  /  between  which  the  solution  lies. 

After  the  first  operation  the  convergence  is  satisfactory,  the  curve  '  sum  =  F{t)'  being  flat 
in  character.     The  result  carries  with  it  the  class  of  triangle. 

Conceivably  this  process  could  be  automatically  carried  out  by  a  machine. 

Parallel  bars  set  at  distances  proportional  to  k,  I,  m  and  kept  parallel  by  a  parallelogram 
linkage,  intersect  fixed  curves  patterned  to  equation  (20).  A  stretched  string  with  its  length 
greater  than  the  sum  in  question  by  a  constant  would  reach  a  mark  as  the  solution  is 
attained. 

It  is  to  be  noted  that  this  method  avoids  the  elimination. 

10.      MULTIPLE   POINTS 

Considering  the  fundamental  equations  in  the  form: 

Fa=a^  —  2a''-{-{i-{-2kt)a—kt  =  o  ,    , 

Fb  =  o,  Fc  =  o,  a+b+c=i 

li  a,b  ,  c,  k,  I,  m,  t  are  a  consistent  set  of  values,  for  neighboring  values  we  have: 

aF.    ,  aF„,  ,  9F„,  ,    . 


MULTIPLE   POINTS  1 7 

and  two  similar  equations.    If  the  point  is  an  ordinary  point  at  which  moreover  none  of  the 

dp  dp  dp 
partial  derivatives  'a"  gi  g-  vanish,  we  have  three  equations  of  the  form: 

Sa  =  ASt+BBk  ) 

U  =  CU+DU  I  (24) 

hc=EU-\-P^m  \ 

First  suppose  A-\-C-\-E^o  and  solve 

o  =  ^a+hb+hc={A+C+E)U+Bhk+DU-\-Fhm. 

This  gives  one  value  of  8/  and  so  one  solution  in  the  neighborhood  {^k,  U,  &m)  of  {k,  I,  m),  that 
is  of  (a,  ;8). 

Secondly  if 

4a^  — 3a+i  ^  ^' 

there  is  a  double  point  in  the  neighborhood. 

Setting  out  the  complete  increment  equations  after  elimination  of  k,  I,  m: 
^^^_(,      )(,         ),^^_^         (  ).,. 

4a^— 3a+i  (a— i)(4a^— 30+1) 


.(26) 


(fl-i)(4a»-3a+i)  /(4a^-3a+i) 

(a-i)(4a'-3aH-i)  (a-i)(4a^-3a+i) 

6(2g-i)  ^  , 

(o-i)(4a^-3a+i). 
and  two  similar. 

When  2 ; =  o  we  have  by  addition  an  equation  of  the  form : 

40^-30+ 1  •'  ^ 

BZk+DU-irP^m  =  o 
in  which  unless  t  =  o  or  a,  b,c  have  certain  special  values  B,  D,  F  are  finite  both  ways. 
If  o,  /8  and  -7-^  are  assigned  the  ratios  k-\-^k  :  l-{-U  :  m+Bm  are  given. 

Let 

k+Sk     l+Sl     m+8m 

"1^  =  1^  =  -^^  =  ^  (^7) 

Then 

^  ■    Bk+Dl+Fm  „,       „     ,  ,    , 

^^Bk'+W+Fm'^^^i'^  -^^  ^^^-  (^«> 

For  an  (a  P)  arbitrarily  near  a  point  for  which  ^+C+£=o  the  terms  of  the  second  order 
on  substitution  of  these  values  for  8^  U  Sm  and  the  values  given  by  the  first  approximations 
for  Sa  Sb  Sc  give  a  quadratic  for  8t. 

Such  points  then  are  double  points  and  the  locus  a  discriminantal  locus. 

Naming  it 

4a"— 3a+i  ^^' 


1 8  THE  PROBLEM   OE   THE  ANGLE-BISECTORS 

we  recognize  that  T  must  be  a  factor  in  the  discriminant  of  F{a,k;  l,m)=o  and  also  of 
^("■j  ")  ^)  =o,  F{a,  k;  l,m)  =  o  being  the  equation  for  the  side  a  in  terms  of  k,  /,  w  (§  15). 

A  triple-point  locus,  that  is  a  locus  whose  intersections  with  T  =  o  are  triple  points,  can  be 
obtained  by  forming  the  second  derivatives  of  a,  b,  c  with  respect  to  t  and  expressing  the  condi- 
tion that  their  sum  vanishes. 
In  terms  of  a,  b,  c  this  is 

^  _^{a-i){2a-i){^a'-ta+2,)ai 

^-^  (4a-3a-Hi)3 =°  ^^o) 

To  express  this  in  the  y,  z  plane,  however,  requires  a  laborious  elimination,  and,  owing  to  the 
practical  necessity  of  multiplying  up  the  denominators  the  result  (which  aside  from  an  infinite 
value  of  2  is  of  the  fourteenth  order  in  z)  is  not  free  from  extraneities,  in  fact  for  the  point 

both  T  and  5  vanish,  but  the  point  in  fact  is  only  a  double  point.  The  finite  triple  points  will 
be  determined  (§  12)  by  another  method. 

Multiple  points  however  can  occur  when  T  does  not  vanish,     li  k  =  l,  a^b 

and  in  this  formula  interchange  of  (8^,  8/)  in  general  changes  this  value  but  leaves  a-^^, 
/?-|-8/3,  which  are  symmetric  functions  of  ^+8^,  l-\-U,  m+8tn  unchanged.  Hence  8/  has  two 
values  and  the  locus  is  discriminantal  for  F{a-,  a,  /?)  =0  but  not  for  F(a,  k;  l,m)=o. 

The  finite  number  of  points  where  ^  =  /,  a^b,  and  £  =  Dare  given  by  4c3-f-4c'-|-c— i  =0  and 
form  an  apparent  exception  but  as  they  are  discrete  points  on  a  continuous  locus  there  are  still 
two  values  of  8;  for  points  as  close  to  them  as  we  please. 

If  ^  =  /,  a  =  b,  then  B  =  D  and  U  has  only  one  value. 

Naming  the  locus  k  =  l,  a^b  zs,  D:,=o,  we  see  that  for  values  of  a,  j3  satisfying  the  (a,  /3) 
equation  of  the  locus  the  double  root  recurs  three  times. 

For  the  first  two  equations  of  the  system  (2 1)  are  identical  and  if  a^jU^,  a,  are  their  roots  and 
fli,  flj,  c  are  the  sides  of  the  triangle o,-(-a2-|-c=i  and  ai+a2+a3  =  2;  whence  a^=c-{-i  and  if 
we  write  ^  =  i 

I  —  2C  (c-f-l)c"  ^•^    ^ 

Setting  aside  c  =  o  as  not  finite,  we  have  a  cubic  in  c 

2c^{m—i)-\-c^{m+T,)  —mc-\- 1=0. 

For  every  m,  that  is  for  every  a,  /3  on  the  locus,  there  are  three  values  of  c  and  so  three  double 
points.     The  discriminant  of  the  cubic  is 

m{()m^-\-T,?>nC-]-()m-\-2i6)  (33) 

If  m  vanishes  the  double  value  of  c  is  i  and  the  corresponding  triangle  has  sides  in  the  ratios 

I    _        I 

'  ■  1/3  '     1/3  ' 


MULTIPLE   POINTS  1 9 

This  is  an  eight-point  but  13  is  infinite  and  so  it  is  not  to  be  listed  in  the  finite  multiple  points. 
For  the  cubic  factor,  which  is  irreducible,  there  are  three  four-points,  one  with  a  value  of  m 
near  —  5  and  two  complex  values.  These  are  seen  later  to  correspond  to  intersections  of  T  =  o 
andZ>j  =  o. 

For  the  locus  k  =  l,  a  —  b,  the  locus  of  isosceles  triangles,  to  be  named  Z),  =0,  proceeding  as 
above, 

m  =  Y nV-t- \77 N  (34) 

(2c-i)(c+i)'(c-i) 

If  c=  I,  then  m  =  %,  t  =  o,  a,  /8  are  indeterminate.     The  triangle  has  sides  as  i  :  o  :  o. 
There  remain  three  values  of  c  for  every  m  leading  to  isosceles  triangles  given  by 

2im-4.)ci+{sm+8)c'-m  =  o  (35) 

The  discriminant  is 

m(2'jm'+gm+T,2)  (36) 

For  m  =  o  the  double  value  is  c  =  o,  not  finite  while  the  complex  values  of  m  lead  to  two  complex 
triangles  in  which  a  part  of  the  isosceles  solutions  coincide. 
These  are 

^ .  h. .  ■■  I9£i_r'5 .  19T1  -15 .  28±i  -15 

a  :  0  :  c  : : :  : . 

94  94  47 

The  points  are  only  double  points  in  the  general  problem. 

Next  are  to  be  considered  the  points  which  are  singular  in  the  individual  equations  of 

the  system. 

dp 
If  ^  =  o  the  equations  take  the  form 

8^.  =C8t+DU  (37) 

Be  =E8t-\-FSm  I 

which  give 

{C+Eyk'+[2{C+E){DBl+F8m)-A']8t+[{DBl+F&my-BU]  =  o  (38) 

Since  this  quadratic  has  one  root  which  approaches  zero  with  Bk,  Bl,  Bm  as  a  rule  only  an 
ordinary  point  exists  for  the  problem.  If  A '  vanishes,  which  happens  for  a  =  i ,  one  of  the  values 
in  question,  the  point  is  a  double  point  at  least,  but  as  yS  is  infinite  not  among  the  finite  set. 

If  /  =  w,  6=t=c  there  is  a  double  point  as  before  on  the  locus  i)j  =  0. 

If  ^Fa/^a  and  ^Fb/^b  both  vanish  and  a  =  b,Bt  is  given  by  a  quartic,  three  of  whose  roots 
approach  zero  with  8^,  8/,  Bm.    This  gives  triple  points  for  the  triangles 

n  ■  h-  r--  3+V  -7  .  3  +  1    -7  .  i-V  -7 
a.o.c.         8         •         8         •         4 

and  the  conjugate  values  which  are  on  r  =  o  and  a  :  6  :  c  : :  i  :  i  :  —  i  for  which  a  =  oo  ,^=00  . 
If  both  the  i)artial  derivatives  vanish  and  a=t=6,  8<  is  given  by  a  quartic  two  of  whose  roots 
approach  zero. 


20  THE   PROBLEM   OF   THE   ANGLE-BISECTORS 


This  applies  to  the  triangles 


a:b:c:^-^±\^:^-^^::^:i 


: :  I 


: :  I 


8 

(3+r  -7) .  (-5-1  -7) 
8  ■  8 

■i3-v-y) .  (-5.+V  -7) 


which  are  double  points  merely,  though  they  formally  satisfy  T  =  o  and  5  =  0. 

Three  of  the  partial  derivatives  cannot  vanish  together  as  a-\-b-{-c=i   and  the  only 

('?±l/  — 7) 
permissible  values  are  i, 3 — —  . 

If  a  partial  derivative  is  infinite  a  side  has  the  value  f  from  which  k  is  infinite  and  /8. 

Finally  none  of  the  special  values  give  finite  multiple  points  not  on  one  or  other  of  the  dis- 
criminantal  loci  T  =  o,  D2  =  o. 

The  nature  of  this  investigation  naturally  leaves  a  doubt  as  to  its  sufficiency  and  may 
be  regarded  as  a  mere  reconnaissance  which  serves  the  purpose  of  gathering  material  to  lighten 
the  labor  of  a  conclusive  determination  subsequently  undertaken  (§  12). 

II.      THE   TRANSFORMATIONS 

Beginning  with  the  configuration  in  the  (a,  b,  c)  plane,  take  as  reference  triangle  an  equi- 
lateral triangle  and  as  the  homogeneity  relation  a-\-b-{-c=  i.  Every  point  in  the  plane  repre- 
sents an  analytic  triangle  with  unit  perimeter.  Each  such  triangle  has  a  sixfold  representation 
corresponding  to  the  permutations  of  a,  b,  c,  the  six  points  having  a  sixfold  central  symmetry. 

The  lines  —a-\-b-{-c  =  o,  a  —  b-\-c  =  o,  a-\-b—c=o  form  a  proper  triangle  dividing  the  plane 
into  compartments.  Inside,  all  the  triangles  represented  have  positive  sides  and  are  possible 
triangles.  In  the  regions  outside  one  of  the  lines  the  represented  triangles  are  impossible. 
Outside  two  of  the  lines  the  triangles  can  be  constructed,  one  side  being  considered  negative, 
and  the  bisectors  meeting  this  side  being  external.  The  sixth  part  of  the  plane  shown  in  Fig.  5 
has  the  regions  ©,  o,  00,  000,  5,  6,  7,  as  subdivisions  of  the  region  of  impossible  triangles  with 
real  sides  while  the  regions  1,2,3, 4,  7',  8,  9, 10,  include  all  the  real  possible  triangles,  the  region 
I  alone  representing  the  real  possible  triangles  with  positive  sides,  that  is  triangles  in  the 
ordinary  sense. 

In  the  regions  5,  6,  7,  the  angle-bisectors  are  real  though  the  triangles  are  impossible,  while 
in  ® ,  o,  00,  000,  the  bisectors  are  pure  imaginary  quantities. 

Considering  in  detail  the  discriminantal  curves,  we  have  first 

^^^a(i-a)(i-2a)^^ 

4a^-3a+i  ^' 

4y'—2oyz-\-40Z^—y-\-z  =  o  (39) 


In  the  y,  z  plane  the  form  is 


As  this  is  an  ellipse  every  real  (y,  2)  is  finite  and  so  every  real  (a,  b,  c)  is  finite  and  the  curve 
is  closed  in  the  {a,  b,  c)  plane. "  It  touches  the  sides  of  the  reference  triangle  at  the  mid-points 
and  passes  through  the  vertices  perpendicular  to  the  medians. 

From  the  summation  form  it  has  no  point  such  that  a,  b,  c  are  all  positive  and  less  than  J. 


THE   TRANSFORMATIONS 


21 


22  THE   PROBLEM   OF   THE   ANGLE-BISECTORS 

By  writing  c=i  —  a—b,  a-\-b=2i,  a—b=2r]  a  sextic  is  obtained  which  contains  i)  only 
as  rf,  r}*  from  which  any  number  of  points  are  easily  obtained  and  multiplied  by  the  sixfold 
symmetry. 

The  curve  is  a  trefoil  not  entering  the  reference  triangle  and  crossing  the  compartments 
o  and  oo  in  the  fundamental  region  (Figs.  4,  $). 

The  set  of  lines  'w{a—b)  =  o  while  not  discriminantal  are  so  transformed  as  to  lie  on  D{a,  p) 
which  is  discriminantal. 

This  locus  of  isosceles  triangles  has  been  named  Di. 

The  locus  of  equal  bisectors  for  scalene  triangles  k  =  l,  a^b  has  for  its  equation 

2ab{a-\-b)-  {d'+Sab-\-b'')-\-2{a-\-b)- 1  =  0  (40) 

and  is  to  be  taken  with  two  other  curves  obtained  by  cyclic  interchange  to  constitute  the  com- 
plete locus 

A(a,  6,  c)  =  o  (41) 

Each  constituent  has  three  real  asymptotes 

a=h,    b=i,    c  =  \ 

with  cusps  at  the  infinite  point,  and  no  finite  singularity.  Two  branches  touch  the  sides  of 
the  reference  triangle  at  the  vertices,  and  externally. 

In  the  fundamental  region  (Fig.  5)  the  various  branches  separate  the  regions  5  and  6, 
10  and  9,  8  and  9,  4  and  3,  3  and  2,  6  and  7. 

Other  discriminantal  lines  are  "n"(a-)-6)  =  o,  which  leads  to  j8=  co  and  'w{a-\-b—c)  =  o  giving 
)8=o.  The  loci  corresponding  to  zero  and  infinite  values  of  a  are  too  compHcated  for  their 
utility  at  this  stage  and  are  added  later. 

In  the  fundamental  region  a-\-b  =  o  separates  00  and  000;  4  and  9;  3  and  8;  2  and  7'. 
Thelinea-|-6—c=o  separates  I  and  ®;  10  and  o;  9  and  00;  4  and  000. 

To  the  (a,  b,  c)  plane  and  this  collection  of  curves,  following  the  process  of  the  elimination, 
is  now  applied  the  "elementary  symmetric  function  transformation," 

a-\-b-\-c      =x, 

ab+bc+ca=y,  I     (3) 

abc  =z. 

Passing  to  a  rectangular  system  by  writing  x—i  and  transforming  the  various  curves  the  con- 
figuration in  the  y,  z  plane  can  be  set  out.  The  transformation  is  point  for  point  between  the 
fundamental  region  in  the  (a,  b,  c)  plane  and  the  region  within  the  discriminantal  curve  in  the 
(y,  z)  plane.    This  curve  is 

D,=  (a-b){b-c)ic-a)  \ 

in  {a,  b,  c)  and  /  (42) 

A  =  4y'—3'^—i8yz-f  272^+42  =  0  ) 

in  the  (y,  s)  plane. 

The  region  of  the  (y,  2)  plane  without  this  curve  represents  complex  points  in  (a,  b,  c)  the 
ratios  of  the  "sides"  being  of  the  form 

a  :  b  :  c  : :  i  :  p+q]/—i  :  p—q\/—i. 


THE   TRANSFORMATIONS 


23 


24  THE   PROBLEM   OF   THE  ANGLE-BISECTORS 

Six  such  points  corresponding  to  the  six  permutations  of  a,  b,  c  are  represented  by  each 
point  in  (y,  z). 

We  may  say  shortly  that  the  transformation  brings  up  from  the  complex  regions  those 
triangles  whose  sides  have  ratios  which  while  not  real  are  the  roots  of  a  cubic  equation  with 
real  coefficients  (Fig.  6). 

The  curve  Di{y,  s)  =  o  is  a  semi-cubical  parabola  and  can  be  written  as 

4(33'-0^+(9>'-272-2)^  =  o  (43) 

The  cusp  (y=h^  =  'ii)  corresponds  to  the  point  A  in  the  {a,h,c)  plane  {a=h  =  c=^). 

The  tangent  at  the  cusp  is  y'  =  32'.  The  curve  cuts  the  y  axis  2^tF  {y=\,z  =  o){a  =  o,b  =  c=\) 
and  has  ordinary  contact  with  it  at  the  origin  corresponding  to  the  vertex  of  the  reference  tri- 
angle in  the  fundamental  region  in  (a,  b,  c).     We  locate  also  the  points 

Z  ;  a  :  6  :  c  : :  2  :  I  :  I,    y=  {(,,    2=  3V  \ 

H  ;  a  :  b  :  c  ::  —1  :  1  :  1,     y=  —  i,     z=  —  i  f 

v.,.j,.,..  (3--/I7)  ■  (i+F  17)  .  (i  +  v  17)      .._(51    17-19)      ■_(5-3l/i7)(  (44) 
B,a.b.c..  -^  .—- g  . ^ ,    y- ,    z- ^^ 

C  the  conjugate  of  B. 

B  and  C  occur  in  the  solution  of  the  equilateral  triangle  case  and  are  on  both  Dt  =  o  and  Z)j  =  o. 
The  locus  1)2=0  becomes  in  the  (y,  z)  plane 

(y-22)=  (y-h22)-42(y-22)-f2  =  o    •  (45) 

This  cubic  has  asymptotes 

y— 22  — i  =  o    with  a  cusp  at  infinity, 

y+  22-|- 1  =  0    with  the  third  intersection  at  2  =  —  J. 

At  the  origin  the  inflexion  y^+2  =  o  gives  a  three-point  contact  with  Z),  =  o.  Z?,  and  A  also 
touch  at  B  and  C.  They  cut  at  H  and  after  entering  at  H  the  real  region  (A  >o),  A  remains 
continuously  within  Di  to  the  infinite  line.  The  part  of  D^  without  A  is  of  course  not  repre- 
sented in  the  real  (a,  b,  c)  plane. 

The  locus     r=o  in  the  (y,  2)  plane  is 

4y'—2oyz-\-4oz'—y-\-z  =  o  (39) 

This  ellipse  meets  Di  and  A  at  the  origin  and  meets  D,  at  F.  It  lies  partly  within  and  partly 
without  the  real  region  (Z),  >o).  The  cusp  of  D,  the  point  A  is  within  the  ellipse.  T=o  also 
touches  y— 2  =  0  at  the  origin. 

■    The  locus  1T(a-|-6)  =  o  becomes  y— 2  =  0.  (46) 

The  locus  ■IT(a-|-&—c)  =  o  becomes  4y—8z— 1  =  0  and  touches  Z?i  at  F.  (47) 

The  locus  a6c  =  o  becomes  2  =  0.  (48) 

Since  the  transformation  of  the  fundamental  region  is  point  for  point  there  is  no  trouble  in 
transferring  the  compartment  markings  from  the  (a,  b,  c)  plane.  For  new  compartments  we 
have,  introducing  as  dividing  curves, 


THE   TRANSFORMATIONS 


25 


26  THE  PROBLEM   OF   THE   ANGLE-BISECTORS 

Fi  .  .  4y'-8yz+z-y  =  o     (a  =  o,  ^=o)  (49) 

Ci    .    .    y^— 23/22 -|-3-yz—SS'  —  Z  =  0      (a=co)  (50) 

•W,  :  A<o,  y-2<o,  Z/',<o 
Ifj  :  A<o,  3»— z<o,  /7,  >o 
XXi  :  A<o,  y— z>o,  r<o,  3;— 23<o 
XXj  :  A<o,  y— 2>o,  r>o,  y— 2z<o 
XXX' :  D,<o,  y-2z>o  T<o,  4y-8z-i<o 
XXX  :  A<o,  )'-2z>o,  T>o,  4>'-8z-i<o 
X,  :  Di<o,  4y—8z—i>o,  T>o 

X2  :  Di<o,  4>'— 8z— i>o,  T<o,  z>o  /(Si) 

Pi  :  Z>,<o,  z<o,  D2>o,  Ci>o,  /7,<o 
Pj  :  Di<o,  z<o,  A>o,  Ci>o,  Hi<o 
Pj  :  D2<o,  2<o,  I>2<o,  Oo,  Hi<o 
Ml  :  A<o,  z<o,  A<o,  Oo,  Hi<o 
Pi'  :  A<o,  z<o,  A>o,  Ci<o,  Zr,>o 
P3'  :  A<o,  2<o,  A<o,  Ci<o,  £ri.>o 
W^i'  :  Z>i<o,  2<o,  A<o,  Ci>o,  ffi>o 

In  tracing  the  intersections  of  the  curves  Ci  and  Hi  with  themselves  and  the  other  curves 
of  the  (y,  z)  plane  we  notice  that  Ci  passes  the  origin  with  an  inflexion  yi—z  =  o  and  so  goes 
from  ®  to  000.  Since  the  asymptote  4^— 82+1  =  0  lies  entirely  in  000  in  the  third  quadrant 
and  the  curve  crosses  it  at  z=  —  gV  ^^^  does  not  cross  the  y  axis,  it  must  remain  in  this  com- 
partment to  infinity.  Leaving  the  origin  in  the  first  quadrant  the  curve  does  not  cross  FZ 
(431—82—1  =  0)  but  crossing  A  enters  XXX.  As  it  reaches  y=2z  at  J  (2,  i)  which  is  outside 
T  it  must  next  cross  T,  pass  through  XXX',  and  enter  XXi  at  /  and  remain  in  this  region  to 
infinity.  This  serpentine  branch  of  Ci  cuts  //,  at  the  origin  and  in  two  other  real  points,  one 
in  XXX  and  one  in  XXX'.  The  other  branch  is  parabolic,  approximating  y'+$z  =  o,  and  lies 
in  the  third  and  fourth  quadrants.  It  crosses  the  z  axis  at  z=  —  ^,  outside  A  at  whose  cross- 
ing z=— 5*7,  and  meets  A  at  H  y=  —  i,  2=  — i  where  the  curves  have  ordinary  contact. 
There  is  no  further  intersection  in  these  quadrants. 

The  existence  of  the  compartments  named  is  thus  proved. 

In  transforming  to  the  (<#>,  t)  plane  we  have  the  birational  transformation 

.^(«^-T+i)  ^  _     <t>(<t>-r) 


with  the  reverse 


42(y-22) 


(y-z)  (43'-8z-i)  '  (y-z)  (43;-82-i) 


(9) 


Real  (y,  z)'s  give  real  (<^,  t)'s  and  conversely. 

Complex  (3/,  z)'s  give  real  (<^,  t)'s  only  for  3'=2s;  <^  =  o,  t=  — i.  The  whole  linear  locus 
31-22  =  0  complex  as  well  as  real  is  thus  represented  at  the  point  4>  =  o,  t=  — i,  but  no  other 
complex  points  in  the  (y,  z)  plane  give  real  points  in  (</>,  t).  Other  singular  lines  of  the  trans- 
formation are 


THE   TRANSFORMATIONS  27 

z  =  o  which  becomes  the  point  ^=0,  t=o  but  as  Limit  -=y  the  linear  elements  at  the  point 
represent  the  various  values  oi  y.     y—z  =  o  gives   <^  and  t  infinite  with  Limit   -=—42. 

T 

4y—8z—i  =  o  also  gives  <l>  and  t  infinite  but  Limit  -  =  i  unless  y  and  z  are  infinite  when  ^  —  t  =  i . 

T 

As  before  noted  this  line  <^  — t—  i  =0  occupies  a  unique  position  inasmuch  as  to  every  point  on 
it  corresponds  the  same  triangle,  namely  the  limit  of 

a  -.b  :  c::i  :  p  :  -/>+5 
when  p  is  infinite. 

The  line  y—2z  =  o  gives  <^  =  o,  t=  — i  and  has  in  the  limit  — ; — =  ,  assigning  real 

T+i     42—1  "      " 

linear  elements  to  real  (y,  2)'s,  complex  to  complex. 

The  boundary  curves  in  the  {<l>,  t)  plane  are  (  Fig.  7) : 

D,=  {<t>-ry{<t>-4ry+{<t>-r)  (^^^+28<t>T- S27^)+{<I>-t)  (3,^- i6t)  +  <^=o  (52) 

The  asymptotes  are 

^        <^— T=o,  intersecting  also  at  (o,  o) 

<I>—T=l,  intersecting  also  at  (VV)  ■^^)' 

Corresponding  to  the  factors  (<^— 47)^  is  a  paraboHc  asymptote 

3(<^-4t)»+i28t=o. 

At  <^=  —  I,  T  =  o  is  an  inflexion  <^'3=  54T'  which  has  four-point  contact  with  the  curve  /8=  — 

T 

=  54  at  this  point.     At  <^  =  o,  t=  —  i  is  a  conjugate  point.    The  point  A  is  represented  by  a 
cusp  at  <^=i,  T=-V.    At  B  (<^=  -^MillLJl)  ^  ^^_(45+in   i7)\  ^^^  ^^^^^  j^^g  ^.^^jj^^^,,. 

with  D2  and  also  at  C  the  conjugate  of  J5  in  (abc).    The  point  .ff  becomes  the  infinite  point  on 
<^— 4r=o,  the  axis  of  the  parabolic  branch. 

The  locus  D2  is  '  • 

(<t>-T)  (<^-4t)  (3,^_4t)+<^^  =  o  (53) 

Its  asymptotes  are 

<l>—  T-j-|  =  o  with  intersection  at  <^=— 5,  t=o 

<^—4T+§  =  o  with  intersection  at  <^  =  ^^,  t=^ 

3<^—4T— 2  =  0  with  intersection  at<^=2,  t=i 

At  the  origin  is  a  cusp  <^^=  i6t3. 

The  contacts  of  the  curve  with  D,  have  been  noted.  At  <^=2,  t=i,  which  is  an  infinite 
point  in  ia,b,c)  and  {y,z),  D^  touches  <^— t— 1  =  0,  the  line  which  also  falls  on  D{a,  fi)  in 
the  (o,  /8)  plane. 

The  locus  r= o  is  a  hyperbola 

t(6<^-i)-(6.^'-3<^+i)  =  o  (54) 

the  asjanptotes  being 


THE   TRANSFORMATIONS  29 

The  complete  representatives  of  £>,  and  D^  are  the  irreducible  factors  above  set  out  with 
the  addition  of  <^  =  o  in  both  cases  and  also  </>— t=o  in  the  case  of  Di.  These  factors  which 
vanish  with  a  may  be  properly  set  aside  and  treated  with  the  zero  and  infinite  values  of  a  and 
P  as  nonfinite  discriminantal  factors. 

The  {<i>,  t)  representatives  of  z  =  o,  y—z  =  o,  4^—83—1  =  0  have  been  discussed  (p.  27), 
the  curve  Hi  becomes  <^+ 1  =  0.     The  curve  Ci  (a  =  00 )  becomes 

l6T4+T3(-40<^+l6)  +  T^(33<^^-28<^)  +  r(-IO<^+IO<^^)  +  «^»(<^+l)'=  \  , 

The  asymptotes  parallel  to  <^— t  =  o  are  complex  and  the  infinite  point  on  this  line  is  a 
conjugate  point.  The  infinite  branch  corresponding  to  (<^— 4r)2  is  parabolic.  As  in  {y,  z) 
the  curve  has  two  separate  branches,  and  therefore  at  least  a  square  root  must  be  used  in  express- 
ing the  points.     This  is  suflScient,  for  writing 

<i>—T=x,    <i>—^T  =  y 
we  have 

y  _iia;+2±l'— i44a;5+ii7x'+36a; 
,  2x  gx'—  2X-\- 1 

The  quadratic  denominator  is  essentially  positive  and  so  y  is  infinite  only  for  infinite  a;'s, 
that  is  on  the  parabolic  branch. 

From  the  radical,  limits  for  the  branches  are  obtained. 

The  closed  branch  has  a  cusp  at  the  origin,  i6r3+^'  =  o. 

The  open  branch  touches  t=o  and  so  A  at  P(— i,  o). 

The  closed  branch  has  three-point  contact  at  the  origin  with  Z),,  and  Z),  is  closer  to  the 
T  axis  than  d  for  points  in  the  third  quadrant  near  the  origin.  The  remaining  intersections 
with  Z>i  are  the  infinite  points  on  <^— t  =  o,  and  <^— 4t=o;  a  two-point  contact  at /(o,—  i) 
which  is  a  conjugate  point  on  Z?,,  and  a  single  intersection  approximately  at  <^=— .27, 
T=  —  .  15,  which  is  better  determined  by  the  rational  value  for  /3,  —  \'.  There  are  also  two 
complex  intersections. 

The  curve  Ci  (0=  00)  meets  Di  =  o  at  the  origin,  at  infinity  on  <t>—T=o,  at  infinity  on 
<A— 4r  =  o,  at  apoint  where/3=  —  Y  (<^=— .4653  .  .  .  ,  t=  — .0226)  and  two  complex  points 
which  with  this  are  the  roots  of  an  irreducible  cubic.  At  these  three  points  the  curves  have  an 
ordinary  contact. 

The  locus  p= =  — V  passes  successively  through  the  cuts  of  a=  00  and  Dj,  A, 

T 

<^— T— 1  =  0.     The  latter  point  is  (-|-2>""5)- 

The  cuts  of  a  =  00  Vith  <^-|- 1  =  o,  <^  =  o  loci  for  which  a  =  o  give  as  points  where  a  must  be 
determined  by  the  direction  of  approach, 

/  :  (0,0),    J  :  (o,  —  i)  for  <^  =  o 
and  the  points 

T=o,     —.609.    .   ,     —1.245.    .   ,     —1.646.    .    .  for<^=  — I. 

To  complete  the  essential  features  of  the  diagram  we  notice  that  Z),  has  a  closer  contact 
with  the  T  axis  at  the  origin  than  D,. 


30  THE  PROBLEM   OF   THE  ANGLE-BISECTORS 

The  line  <^=  — i  meets  D,  at  t=  — 1.014,  meets  T  at  t= 

-  ■'^2,2,-  ■  ■ 

These  points  enable  the  ordering  of  certain  compartments  to  be  made  clear. 

The  (<^,  t)  plane  is  covered  as  by  a  single  sheet  of  the  (y,  2)  plane  stretched  but  not  folded. 
Continuity  is  preserved  except  for  the  singular  points  and  lines  of  the  transformation,  which 
become  lines  and  points  respectively,  and  except  that  as  regards  the  infinite  values  the  usual 
conventions  of  the  projective  plane  are  to  be  observed.  It  is  convenient  to  locate  some  special 
points: 

A  :  <I>  =  -1,     r=V-  _  ^ 

^  ■  0_      (13+31/17)       ^_      (45+111/17) 

4  '  16 

C  :  conjugate  to  B 
D  :  <f>=2,    T=  I 
£:3<^  =  4T=oo 

F:4>=T='X)   on  <^— T=o    i.e.,  on  £>i 
G:<^  =  T=oo  on  <^=r+|  =  o    i.e.,  on  D, 

^:<^  =  4r=oo  /(S6) 

/  :  <f>=T=o 
J  :  <^  =  o,    T=  —  I 

L  :  <^  =  3i/  5—  I,    T=  fK  5  and  M  its  conjugate 
^'•4'=  5,    T=4  :  Z,  if,  iV  occur  in  the  case  of  a=4,    ^9=54 
P:<^=-i,    T=o    a  =  4,    /3=S4 
PF  :  <^=2.629  .  .  .     r=,886...     a  =  4,     ^=54 
Z  :  <I>—T=  J,     <^=  CO   on  an  asymptote  of  A. 

The  region  (i)  is  within  the  cusp  of  A  at  A  and  reaches  to  the  infinite  with  F  and  Z  as 
limits  of  the  branches. 

The  region  (4)  is  identified  by  means  of  H,  D,  and  W,  which  are  on  its  boundary,  while 
E  is  not. 

The  region  (3)  is  bounded  entirely  by  D2  and  infinite  points  and  reaches  D,  H,  and  E 
and  is  then  inside  the  branch  of  A  in  the  first  quadrant  of  (<^,  t). 

The  region  (2)  is  located  by  E  and  N  and  by  its  separation  from  (3)  by  A- 

The  regions  (5),  (6),  (7)  all  reach  C  and  (5)  is  bounded  entirely  by  D,  and  A;  (6)  reaches 
/  with  (5),  while  (7)  does  not.  (7')  which  in  iy,z)  has  continuity  with  (7)  through  infinity, 
has  in  (<^,  t)  continuity  through  the  point  P  and  joins  (4)  in  the  infinite  regions  in  (^,  t)  just 
as  it  joins  (4)  in  {y,z)  along  y—z  =  o. 

The  regions  (8),  (9),  (10)  have  contact  of  their  boundaries  at  B,  and  (9),  being  entirely 
bounded  by  A,  is  the  inside  region  in  (<^,  t).  (8)  reaches  H  in  (y,  z)  and  so  must  belong  to  the 
parabolic  branch  of  D2  and  be  the  upper  one  of  the  three  regions. 

Of  the  regions  with  real  sides  and  imaginary  bisectors  0  which  in  (y,  z)  reaches  F  and  Z 
and  is  bounded  by  A  and  2  =  o  in  (<^,  t)  is  bounded  by  Di  and  <;(>  —  t  =  o  and  reaches  I. 

o  bounded  in  (y,  2)  by  A,  2=  o  and  43/—  82—  i  =  o  in  (<^,  t)  lies  in  the  first  quadrant  between 
A  and  <t>—T  =  o.    00  is  continuous  with  o  through  A-    000  in  (y,  2)  joins  00  along  y— 2=0 

which  involves  <^=  00 ,  t=  00  with  -=  —  42  and  2  runs  from  zero  to  —  j  so  in  (<^,  t)  these  regions 


THE   TRANSFORMATIONS  3 1 

are  continuous  through  infinity  in  the  second  and  sixth  octants.  000  reaches  /  by  passing 
through  /  a  singular  point.  In  (y,  z)  000  is  continuous  with  XXX'  through  infinity,  in  {<t>,  t) 
through  the  line  <^— r— 1  =  0. 

The  regions  outside  Z),,  that  is  corresponding  to  complex  sides,  are  to  be  identified  as  follows: 

The  line  Hi  =  o  becomes  <^+i  =  o  and  the  origin  in  (y,  2)  has  linear  elements  which  cover 
<^  =  o,  T  >o  hence  M2  bounded  also  by  A  and  reaching  C  is  identified. 

For  the  region  W,,  the  asymptote  oi  H^  y  =  i  reaches  infinity  at  a  point  which  becomes  P 
and  as  Wi  reaches  y  =  z  for  all  positive  values  of  z  in  the  {<i>,  t)  plane  it  must  reach  all  infinite 
(<l>,  r)'s  which  are  the  limits  of  <t>=  — 42T  and  so  occupies  the  remainder  of  the  second  quad- 
rant after  M2  is  removed.  The  same  argument  locates  XXi  and  so  XX2  by  crossing  T  but  not 
<^  =  o.  The  regions  XXX  and  XXX'  can  now  be  reached  through  /,  the  linear  boundaries 
in  (y,  z)  being  replaced  by  the  collection  of  linear  elements  at  /,  which  is  a  singular  point  of 
the  transformation. 

In  the  (<^,  t)  plane  it  is  convenient  to  subdivide  these  compartments  by  means  of  the 
lines  Hi  and  Ci  representing  zero  and  infinite  values  of  a. 

Xi  is  identified  by  A ,  and  the  cusp  on  Dt  and  X2  by  crossing  T. 

©  is  reached  from  XXX  by  crossing  Z>i. 

Ml  is  reached  from  7'  by  crossing  A,  thence  crossing  C,  (a=  co)  we  arrive  at  P^;  across 
Hi  (a  =  o)  to  Pj',  across  a=  co  to  W,',  across  D2  to  W2,  across  a=  00  to  P/,  across  a  =  o  to  P„ 
across  a=  00  to  P2  (see  Fig.  8). 

It  is  convenient  to  subdivide  also  the  regions  ©  by  <^=  —  i,  00  by  T,  000  at  /  and  again 
by  a  =  00  . 

In  the  (a,  /8)  plane  the  discriminantal  loci  to  be  traced  are: 

£)(a,  /3)  the  representative  of  £>,,  D2  and  the  line  <t>  —  T— 1  =  0 
T(a,  /3)  =  o,  the  axes  and  the  infinite  boundary  (Fig.  9). 

On  account  of  the  single  value  of  a  for  given  j8,  <j  and  the  connection  with  a  model  of  the 
surface  F{(t,  a,  /3)  =  o  the  a  axis  is  taken  vertical.  To  trace  T  =  o  which  is  a  rational  curve  in 
(<^,  t)  the  parametric  representation,  obtained  by  substituting  for  t  its  rational  expression  in 

<tt,  namely  — t^^ — r —  in  the  (</>,  t)  expressions  for  a  /3  (10). 
(6<^— I) 

This  gives 

a=  ('^+i)'(6<^-i>(2'»-i)  \ 

(57; 


„     («^+iW6<^-i) 


S 


Letting  <t>  range  over  all  real  numbers  we  have  a  real  branch.  It  is  proper  however  to 
inquire  whether  any  other  branch  exists  by  which  complex  (<^,  t)'s  are  represented  by  real 
(«,  /3)'s. 

The  general  condition  that  such  may  be  the  case  is  : 

If  a=f(<f,)=f{a+bi)  =  A+Bi 
and  ^=g{<f>)  =  g(a+bi)  =  C+Di, 

B  and  D  have  a  common  factor  other  than  b. 


I  (58) 


32 


THE  PROBLEM   OF   THE  ANGLE-BISECTORS 


In  this  case  P  is  real  if  6  =  o  or  if 

3(4a-i)[664-3i,^(i2a^+i7a+5)+(6a-i)(a+i)3] 
-|-[6J^-(6a^-3a+i)][-J^(24a+i7)+3(8a+i)(a+i)»]  =  o 

The  coefficient  of  h*  is  (—72^—120)  and  a  set  of  terms  free  from  b  exists. 
For  a=i  the  expression  becomes 

—  192Z)''— 1066^+72 
which  does  not  reduce,  and  hence  the  general  expression  does  not  reduce. 

F.J  8, 


oL  ■f\>Mte 


d-* 


c^-^ 


;      r 

/ 

11 
(SJ. 

0 

/' 
0-"/ 

+- 

0,    +  + 

"  -4- 

»l 

/ 
/ 
/ 
/ 

/ 

"^ 

^^-"-^ 

"  i^"'"~'7 

\ 

t 

./"'^^  -4 

J^--  / 

\ 

^.^'^ 

/.y^'r- 

—  h 

/ 

/ 

+; 

>' 

• 
• 

+  - 

(59) 


For  a=o,  &'  =  f ,  y3  is  real  but  the  corresponding  expression  for  a  is 

[432&«+2886''-(-ios6"-  i][4326i+4286^- 102]-  [4326"+ 1646'-  i8][3i26'- 15]     • 

when  a  =  o,  and  this  is  not  zero  when  h''  =  \. 

Hence  there  is  no  such  common  factor  and  r(a,  /8)  =  o  is  a  unicursal  curve  and  <^  is  a  proper 
parameter. 

It  is  moreover  in  i  :  i  correspondence  with  the  hyperbola  in  the  (<;^,  t)  plane,  and  the  facts 
discovered  there  may  be  utilized  in  tracing. 

The  curve  T  meets  a  =  00  in  one  point  only,  for  at  /,  which  is  apparently  an  intersection, 
the  linear  elements  differ  and  /  is  singular. 

This  gives  a  single  asymptote  /8=  .0164  .    .    .  approximately. 

j8  is  infinite  for  no  real  finite  ^  but  for  <^=°o,  a=oo,  ^=00. 

This  gives  a  parabolic  branch  with  /3=a»  as  parabolic  asymptote. 


ON 


34  THE  PROBLEM  OF   THE   ANGLE-BISECTORS 

For  <^=  —  I,  a  =  o,  /8=o  the  character  being  given  by 

54  iy3a3=  2^.33.  77^2     a  cusp. 

At  <l>=i,  tt  =  o,  ^=0 

2''.3''.;8-'-f  77.0=0     an  inflexion. 

Since  the  curve  crosses  D2  at  the  real  four-point  it  must  touch  D{a,  j8).    This  occurs  at 
a=  — .99678.    .     /3=5.3542  .        .     approximately. 

For<^  =  i;  a=o,y8=-V-. 

For  4>=o;  a=iV,  /3=-i. 

For  /8=  —  I  <t>'  =  o  or  <^  is  complex.  This  point  which  corresponds  to  /  is  then  a  turning- 
point  for  j8.     For  j8<  —  i  all  the  values  of  <l>  are  complex. 

There  is  a  maximum  value  for  a  at  <^=  —  33  .  .  ,  a=  .66  .  .  ,  ^=  —  .33,  and  a  maxi- 
mum value  at  "^=.402  .  .  ;  a=  — i.oi  .  .  ,  18=5.45  .  .  ,  and  a  maximum  value  at 
<^=  — 1.58  .  .  ;  a=— .8307  .  ,  /8=.3oi2  .  .  .  The  curve  meets  D  only  at  the  origin 
and  the  four-point. 

These  and  the  negative  facts  implied  by  omission  are  sufficient  to  establish  the  general 
character  of  the  curve. 

In  the  (<^,  t)  plane  the  curves  (<^-|-i)^  — /8t  =  o  form  a  family  of  cubic  parabolas  with  centers 
at  i'(—  I,  o)  and  can  be  easily  visualized  in  the  diagram.  By  so  doing  it  is  seen  that  the  num- 
bering of  the  regions  i,  2,  3,  4,  5,  6,  7,  8,  9,  10  corresponds  to  the  order  of  magnitude  of  the 
real  roots  of  F(o-,  a,  j8)  =  o. 

Since  A  has  three-point  contact  with  the  curve  /?=  54  of  this  family  at  P,  and  a  here  inde- 
terminate has  the  limit  4  for  this  approach,  the  region  7'  is  seen  to  contain  only  values  of  yS 
which  are  greater  than  54  and  to  be  continuously  joined  to  (7)  along  a=4  in  the  (a,  ^)  plane. 
This  corresponds  to  the  change  of  class  of  the  root  from  (113)  to  (331)  when  a  passes  the 
value  4  (§8). 

Digressing  to  complete  the  comparison  of  the  two  classifications  of  the  real  roots  we  have 
the  set  (122)  [§8]  has  one  negative  side  and  as  the  sum  of  the  positive  sides  is  not  greater  than 
2,  the  negative  side  cannot  be  less  than  —  i.  This  identifies  the  set  of  three  with  (8),  (9),  (10). 
(122)  has  the  greatest  bisector  internal  and  opposite  a, :  hence  j  a,  |  is  the  least  magnitude  among 
the  sides  and  as  it  approaches  zero  the  other  sides  approach  5,  J  which  is  the  triangle  repre- 
sented at  F.  This  identifies  (122)  and  (10).  (221)  can  approach  H  (i,  i,—  i)  and  is  then  (8) : 
(212)  is  (9).  The  set  (231),  (312),  (321)  correspond  to  (2),  (3),  (4),  and  (2)  reaches  the  line 
a+b—c  =  o  or  has  a  side  equal  to  |.  This  must  be  c^  for  if  Oj  or  bi  has  this  value  c,  is  infinite. 
Thus  (2)  is  (312).  (4)  reaches  a+b  =  o  or  one  side  has  the  value  i.  This  must  be  a^.  So 
(4)  is  (231)  and  (3)  is  (321). 

For  the  set  (5),  (6),  (7),  consider  the  approach  to  (i,  o,  o):  only  ai+bi+c^  can  reach  this 
point  with  a  and  b  negative  and  nearly  equal :  (113)  is  (5).  (131)  can  approach  with  a  and  c 
negative  and  very  unequal,  this  is  then  (6).  (311)  is  then  (7),  while  (7')  having  two  positive 
sides  greater  than  i  and  a  negative  side  less  than  —  i  is  (331). 

The  transformation  from  the  (<^,t)  plane  to  the  (a,  j8)  plane  effects  a  10  :  i  correspondence 
and  brings  as  in  general  complex  (<^,  t)'s  into  correspondence  with  real  (a,  /3)'s. 


THE    TRANSFORMATIONS  35 

Beginning  with  the  real  regions  of  the  (<^,  t)  plane  it  is  necessary  to  determine  the  limiting 

values  of  a  and  ^  at  the  points  where  they  become  indeterminate  and  also  for  the  infinite  values 

of  (<^,t)  by  various  paths  of  approach. 

,     ,  .        Am^im—i) 

The  hmit  of  a  for  <^  =  ott=  oo  is -.-. r, . 

{m—iY{m—/^y 

The  limit  of  a  along  (<^+i)^— /3t  =  o  for  finite  )8  as  <#>  approaches  <x  is  o. 

The  limit  of  a  along  4>—mT^  =  o  is  4. 

The  limit  of  «  along  m<\>^—r=o  is  o. 

The  limit  of  a  along  <^(</>—  t)  —  i  =  o  is  4. 

For  finite  points : 

At  the  origin — 

.    ±m{m — i) 
The  limit  of  a  along  <^— »2t=o  is  ; . 

The  limit  of  a  along  A  is  4:  along  A  is  =» . 

At  the  point  P  (— I,  o)— 

The  limit  of  a  for  all  rectilinear  approaches  is  o. 

The  limit  of  o  for  approach  on  any  curve  (<^+i)2— ^t=o  is  4. 

The  limit  of  a  for  approach  on  w(<^+i)^— '■=0  is  -, ; — r  and  the  value  infinity  occurs 

,.        ,  (20W+1) 

only  for  negative  t  s. 

At  the  point  7(o,  —  i) — 

The  limit  of  a  for  approach  on  t—  n<}>+ 1  =  o  is  7 ;  . 

^^  (3-4«) 

The  point  J  then  represents  all  real  points  on  the  line  /3+i  =  o. 

At  the  point  P(—  i,  o) — 

The  limit  of  P  for  approach  not  on  a  ^  curve  is  00  . 

Any  value  of  j8  is  reached  by  approach  on  the  /8  curve  (<^+i)^— /3t=o. 

As  a  preliminary  to  identifying  the  compartments  the  signs  of  a,  j8  may  be  marked  on  the 
(<^,  t)  diagram  (Fig.  8). 

The  discriminantal  loci  in  the  {<!>,  t)  plane  are  A,  T  and  the  loci  giving  zero  and  infinite 
values  to  a  and  /3. 

In  transferring  the  {<!>,  t)  compartments  to  the  (a,  /3)  plane  they  must  be  folded  at  the 
proper  discriminantal  lines  A  and  T.     (Though  D,  falls  on  D[a,  /J]  it  is  not  discriminantal.) 

For  the  other  loci  a  special  inquiry  must  be  made.  For  a=  00  ,  e.g.  the  points  on  the  same 
/3  curve,  close  to  and  on  opposite  sides  of  this  locus,  correspond  toa=+»?,a=— w  and  in  the 
(a,  j8)  plane  are  near  only  in  the  projective  sense :  folding  is  then  not  the  proper  word. 

If  such  a  locus  occurred  with  the  vanishing  or  infinite  factor  entering  with  an  even  exponent, 
folding  would  suit,  but  the  only  one  of  this  class  for  finite  (<^,  t)  has  a  further  characteristic 
which  prevents  the  use  of  the  concept.  Namely,  the  locus  <^+i  =  o  contains  only  points  f6r 
which  a  vanishes  to  the  second  order,  while  /8  vanishes  to  the  third.  The  whole  locus  is  a  sin- 
gular line  all  of  whose  points  find  their  representation  at  the  origin  in  the  (a,  P)  plane.  It  is 
only  possible  to  say  that  the  sheets  are  connected  at  this  point. 


36  THE  PROBLEM   OF   THE   ANGLE-BISECTORS 

For  infinite  (<^,  t)  an  example  can  be  found.  The  region  Wi  is  connected  in  the  (<^,  t)  plane 
with  XXi  by  passage  through  the  infinite  line  along  {<t>+iy— I3t=o  for  any  negative  /S.  In 
both  regions,  however,  a  is  positive  and  as  the  limit  of  a  along  the  ft  curve  is  o  the  order  of  the 
vanishing  factor  must  be  even  and  folding  occurs. 

In  the  same  way  for  positive  /8,  00,  and  oooi  are  folded  on  tt  =  o,  in  the  fourth  quadrant  of 
(a,  /3)  since  a  is  negative  in  both  compartments. 

Starting  with  those  regions  of  the  (<^,  t)  plane  which  fall  on  the  first  quadrant  of  the 
(a,  /8)  plane,  the  region  (i)  inside  the  cusp  of  Di  falls  inside  the  cusp  of  D.  Since  £),  is  not  dis- 
criminantal  Xi  is  continuous  with  (i)  over  the  line  D  up  to  the  parabolic  branch  of  T.  Along 
this  curve  the  points  are  readily  identified  by  the  parametric  value  <l>.  Otherwise  Xi  reaches 
<f>—T=o,  that  is  a  =  o,  and  X2,  which  is  folded  on  X,  at  the  curve  T,  reaches  •r  =  o,  that  is  a  =  4, 
^=00,  while  X,  along  <t>{<t>—T)—i  =  o  reaches  for  infinite  <#>  the  same  point  (Figs.  9  and  7 
and  p.  3S). 

The  line  ^  — t—  1  =  0  falls  as  a  whole  on  D  and  the  part  in  the  first  <t>,  r  quadrant  falls  on 
the  boundaries  of  the  "cusp"  region.  The  point  D  (2,  i)  falls  on  the  cusp  and  the  lower  part 
of  the  line  reaching  t  =  o  reaches  a  =  4,  /8=  «> .     The  upper  part  reaches  a=  00  ,  /3=  00 . 

The  line  D2  also  falls  on  D  and  the  hyperbolic  loop  in  the  first  octant  of  {<t>,  r)  falls  on  the 
boundaries  of  the  "cusp"  region. 

The  three  regions  (2),  (3),  (4)  fold  alternately  and  form  a  sort  of  pleat.  (2)  is  continuous 
with  X2  along  the  upper  branch  in  (a,  /3).  The  upper  branch  of  Z?j  in  (<^,t)  falls  on  the  lower 
branch  of  D  in  (a,  /3)  and  vice  versa.  Next  consider  the  region  P2  which  falls  on  the  first  quad- 
rant of  (a,  yS).  P2  reaches  a  =  o,  /3  =  o  along  (j>=  —  i  and  reaches  1  =  4,  fi=  co  along  <^(<^— t)  — 
1  =  0  for  <^=  00  ,  and  reaches  a=  00,  ^=00  along  the  parabolic  branch  of  A.  It  reaches  a=o,  ft 
any  positive,  along  <f>  —  T—o  and  ^=0,  a  any  positive  at  the  cut  of  a=  c»  ,  a  =  o,  <^=  —  i. 

P2  does  not  contain  T  in  (<^,  t)  and  so  in  (a,  j8)  passes  continuously  over  the  branches  of 
T  without  change.  The  regions  (8),  (9),  (10)  join  Pi  over  A  in  (<^,  t)  and  are  pleated  in  (a,  /3) 
over  the  cusp  region.  The  branch  of  D2  between  (8)  and  (9)  falls  on  the  lower  boundary  of 
the  cusp  region,  for  it  is  continuous  through  infinity  with  the  boundary  of  (2),  (3).  B  falls 
on  the  cusp.  For  the  region  Af,  a>4  and  moreover  a  has  a  greater  value  than  belongs  to  the 
branch  of  Z),  which  separates  Mi  from  (7')-  The  region  M2  reaches  a  =  4,  /3=  54  at  P  by  approach 
on  the  /8  curve  and  so  joins  Mi.  Further  M2  reaches  ^  =  0,  a  any  value  between  o  and  4  by 
approach  at  P  along  parabolas  (p.  35).  Mi  reaches  /3=o,  «  any  value  between  4  and  00  in  the 
same  way.  M2  reaches  a  =  o,  /3  any  positive  at  the  points  along  <^  =  o.  M2  embraces  the 
regions  (5),  (6),  (7)  which  are  pleated  on  the  cusp  region,  C  falling  on  the  cusp,  the  fold  of 
(5)  and  (6)  falling  on  the  upper  boundary  and  the  fold  of  (6)  and  (7)  on  the  lower.  (7)  for 
which  all  points  have  a  not  greater  than  4  is  joined  continuously  to  (7')  for  which  all  points 
are  not  less  than  4. 

Since  Mi  and  M2  do  not  contain  T  in  the  {<i>,  t)  plane  they  pass  over  it  without  change  in 
the  plane  (a,  yS).  With  the  regions  (5),  (6),  (7),  and  (7')  which  are  pleated  on  the  cusp  region 
they  form  a  continuous  covering  of  the  first  quadrant  of  the  (a,  /?)  plane  in  the  same  manner 
that  P2  with  the  pleated  regions  (8),  (9),  (10)  does. 

The  sheet  X2  with  the  regions  (2),  (3),  (4)  pleated  behaves  in  the  same  fashion,  while  the 
sheet  Xi  containing  the  region  (i)  giving  the  internal  solution  is  without  fold  at  the  boundaries 
of  the  cusp  region.  There  remains  in  the  first  quadrant  the  pair  of  regions  XXXi  and  XXXi' 
between  a=  00  and  the  negative  side  of  <^-f-i  =  0  and  separated  by  T.    These  reach  all  positive 


THE   TRANSFORMATIONS  37 

a's  for  j8=o  at  the  indeterminate  points  for  a  (p.  29)  and  folding  on  the  asymptotic  branch  of 
T  cover  the  space  between  this  Hne  and  the  a  axis. 

The  sheets  X,  and  X,  are  folded  at  the  paraboHc  branch  of  T  where  4>>\  and  the  complete 
account  of  the  first  quadrant  of  (a,  /8)  is:  10  real  roots  for  the  "cusp"  region,  4  real  roots 
between  this  and  the  parabolic  branch  of  T,  2  real  roots  between  the  two  branches  of  T,  and 
4  real  roots  between  the  asymptotic  branch  of  T  and  the  a  axis. 

It  is  interesting  to  notice  the  persistence  of  the  root  (i)  for  a  region  of  the  (a,  ^)  plane 
much  more  extensive  than  the  region  where  it  has  an  interpretation  as  a  solution  of  the  prob- 
lem for  real  angle-bisectors. 

Taking  up  the  second  quadrant  of  the  (a,  /3)  plane  the  region  W^  has  a  not  greater  than  4, 
/3  any  negative.  Wi  is  continuous  with  Wi  through  P  and  as  the  /3  curves  with  P  as  origin 
are  Pr=ai  and  the  curve  a  =  00  is  2ot  =  o-%  all  the  /8  curves  for  /8<  o  fall  between  a=  00  and  the 
<^  axis.  For  Wi ,  a  is  not  less  than  4.  The  region  is  bounded  by  A  which  cuts  a=  00  at  a  point 
for  which  j8=  —  ^ ,  and  as  this  value  is  asymptotic  for  Z?(a,  /3)  =  o  and  A  is  discriminantal  the 
curve  /3=  —  V  touches  0=  00  at  the  point  in  question  in  the  (<#>,  t)  plane.  For  j8>  —  V  the 
P  curves  leave  W^'  by  crossing  a=  00  .  So  PF,  and  W^'  together  form  a  sheet  covering  the  second 
quadrant  of  (a,  y8)  up  to  the  line  D.  At  this  line  the  sheet  is  folded  and  returns  from  the  fold 
as  W^-  W2  reaches  a=  00  for  o<  j8<  —  V,  ^nd  reaches  a=o,  /3=o  along  <^+i  =  o,  and  reaches 
j8=oo ,  o<a<4  at  /  (o,  o).  Wi  has  these  latter  values  in  the  infinite  regions.  Wi  and  W2 
only  join  (6)  and  (7)  at  0=4  ^=  00 . 

The  (<^,  t)  regions  XXX^,  XXX2  are  continuous  with  XXXt  and  XXXi  respectively 
through  the  cuts  of  a  =  00  and  </)+i  =  o  where  ji=o  and  a  depends  on  the  path  of  approach. 
These  paths  are  parabolas  touching  a=  00  since  this  is  of  the  first  order  while  <^+ 1  =  o  is  of  the 
second,  and  so  in  passing  these  points  from  one  of  the  regions  to  the  other  a  does  not  become 
infinite.  The  regions  are  also  continuous  with  XX2  and  XXi  respectively  through  J.  XXi 
is  continuous  with  00O2'  over  <^— r— 1  =  0,  the  non-discriminantal  representative  of  D{a,  j8). 
The  whole  set  forms  a  double  sheet  covering  the  second  quadrant  of  (a,  /8)  (with  the  exception 
of  the  part  inside  the  loop  of  T)  and  the  part  of  the  first  quadrant  up  to  the  asymptotic  branch 
of  T,  as  before  mentioned.  It  is  folded  on  T  from  the  cut  of  T  and  a=  00  [asymptotic  point 
of  T  in  (a,  13)]  through  <^=  — i  [cusp  of  T  at  a  =  o,  /3=o],  to  J  (o,  —  i)  [a=  ^\,  /3=  —  i]  and  up 
to  <^=g,  the  asymptote  of  r  in  (</>,  t)  [inflexion  of  r  at  a=o, /8=o]. 

The  sheets  reach  a  =  o,  /3  any  negative  as  follows: 

o>)8>  — I  in  XX2  (J  represents  13— —  i  as  a  whole)  and  in  XX i  at  infinite  points  on  j8 
curves.    They  reach  /3=o,  a  any  positive  at  the  indeterminate  points  discussed  above. 

The  region  inside  the  hyperbolic  branch  of  D{a,  /3)  is  covered  by  00O2'  in  one  sheet  and  the 
part  of  XXX2  between  the  cut  of  A  and  a=  co  and  the  origin  in  the  other. 

The  third  quadrant  in  (a,  /8)  has  as  representatives  in  (<^,  t)  : 

©2  and  XX X;,  continuous  over  £>,. 

00O2  and  XXX/  continuous  through  J,  and  since  <^— t— 1  =  0  is  crossed  at  J  and  also 
between  XXX/  and  000,  there  must  be  added  that  part  of  000,  for  which  <^+ 1  >o.  This  part 
has  o  >  /J  >  —  I  but  joins  0002  for  /8  =  —  i ,  o  any  negative  at  J. 

As  no  discriminantal  lines  occur  we  have  two  separate  sheets.  XXX ^  is  continuous  over 
/3=o  with  XXX3,  and  XXX/  with  XXX/  over  the  same  line. 

P/  and  Pi  are  folded  on  D  and  reach  /3=o,  a  any  negative  at  P  and  the  indeterminate 
point  for  a.    At  the  fold  o  >/3>  —  Y  (Fig.  7).    These  facts  locate  the  regions  in  (a,  /3). 


38  THE  PROBLEM   OF   THE   ANGLE-BISECTORS 

The  fourth  quadrant  has  in  (<^,  t): 

The  second  octant  of  (<^,  t)  having  four  regions  d,  O2,  00,,  0O2  which  meet  at  the  four-point 
and  are  folded  so  as  to  hang  together  at  the  four-point  in  (a,  j3).  They  are  joined  in  pairs  along 
£>(<!,  /8)  for  all  values  of  /8  greater  than  the  four-point  value,  for  lesser  values  the  j8  curves  in 
(^,  t)  meet  D^  in  complex  points. 

For  the  lesser  p's  ooi  and  0O2  are  folded  on  T.  d  and  O2  are  folded  on  the  other  branch  of 
r  up  to  {<i>=r=\)  [j8=  Y",  a=o]  after  which  they  pass  continuously  into  X,  and  X^  which  are 
folded  on  this  parabolic  branch  of  T  in  the  first  quadrant  of  (a,  /8). 

The  fold  of  00,  and  0O2  passes  through  the  origin  in  (a,  /8)  and  is  continuous  with  the  fold 
of  the  sheets  away  from  the  loop  of  T  and  reaches  the  first  quadrant  as  the  fold  along  the  asymp- 
totic branch  of  T. 

The  regions  P,  and  Pj  are  folded  on  D2  for  positive  ;S's  and  reach  a=  00 ,  /3=  00  with  Z)^ 
and  a=  CO  ,  y3  any  positive  on  the  curve  a=  00 .  They  then  cover  all  the  fourth  quadrant  below 
the  branch  of  D.    They  are  continuous  with  P/  and  Pj'  over  the  a  axis. 

The  regions  XXX^  and  XXX^'  are  folded  on  T,  reach  a=  00  ,  /3=  00  along  <^— 't=o  :  a=  00  , 
/3=o  at  <#>4-i  =  o  but  for  a=  00  the  greatest  /3  is  .0164  .    .   the  value  for  the  asymptote  of  T. 

In  (a,  /3)  then  these  sheets  are  folded  on  the  lower  branch  of  T  and  extending  upward  pass 
continuously  into  ®  and  000,  respectively  across  Di  and  <^—t— 1  =  0,  both  of  which  fall  non- 
discriminantally  on  Z)(a,  ^). 

Collecting  the  parts  of  the  plane  which  are  continuously  connected  in  (<^,  t)  as  well  as  in 
(a,  P)  we  have,  setting  aside  the  region  of  ten  real  roots : 

Sheet  A  containing  X,,  d  • 

Sheet  B  containing  X2,  O2 

Sheet  C  containing  Af,,  M2,  ooj,  Wi,  W/ 

Sheet  D  containing  P2,  6,,  ©2,  W^,  XXX^,  XXX^ 

Sheet  E  containing  000,,  0002,  XX^,  XXX,,  XXX,,  XXX/,  XXX/ 

Sheet  F  containing  0002',  XX„  XXX,',  XXX,' 

Sheet  G  containing  P,,  P,' 

Sheet  H  containing  Pj,  Pj' 

Sheet  J   containing  00, 

Sheet  K,  the  tenth  sheet,  is  not  represented  outside  the  region  often  real  roots  (Fig.  10). 

The  phenomena  at  infinity  are: 

i)  a=  CO  J  p  any  finite  value. 

There  are  ten  distinct  finite  roots  except  at  the  asymptotic  lines  of  T  and  D,  any  value  of 
o-  being  reached  for  at  least  two  finite  real  values  of  /8. 

2)   /3=oo,a>4. 
Eight  roots  are  infinite  in  pairs.     For  (T'^  =  mfi  they  are  given  by: 

l6aw'—  40ttm3+33aOT^-|-  (4—  IOa)m-(-  (a  —  4)  =  o. 

Two  roots  are  <^=  i,  and  these  belong  to  the  regions  (5),  (6)  if  /?  approaches  infinity  from 
the  positive  side,  and  to  the  regions  XXX,  and  0002'  if  /3  approaches  on  the  negative  side. 


r 


THE   TRANSFORMATIONS 


39 


The  four  pairs  of  infinite  roots  are  (i)  and  (lo)  :  (2)  and  (9)  :  (3)  and  (8)  :  (4)  and  (7')  for 
approach  with  positive  /?,  while  for  negative  ft  all  approach  in  the  complex  regions. 

The  equation  in  m  has  no  negative  roots  for  a>4. 

3)  fi—  00  ,  a  =  4.    One  root  is  zero  for  any  ;3.     This  is  (7)  or  (7'). 

Three  roots  are  finite.    These  are  easily  determined  from  the  4>,  t  equation  for  0  =  4  when 
/8  is  infinite  and  <i>  finite,  for  then  t=o. 

The  equation  is 

^[(<^-0(3<^-4'-)'+4(<^--^)(3<^-4'-)-<^]  =  o. 

Neglecting  the  indeterminate  solution  t  =  o,  there  are  three  values  for  </> :  o, '-^ ,  ^—^ . 


'% 


AO- 


<^=o  belongs  to  (5),  the  negative  value  to  (6)  and  the  positive  to  (4)  for  positive  /J.  For 
negative  /8  they  fall  in  XXX2,  W^,  000/  respectively. 

The  values  obtained  for  a  (or  <^),  however,  depend  on  the  path  of  approach.  If  /3=  00 
a  =  4  the  roots  are  eight  infinite  and  two  indeterminate,  while  for  approach  along  the  lower 
branch  of  D  whose  asymptote  is  a— 4  =  0  the  root  (6)  and  the  root  (7)  are  continually  equal 
and  attain  the  limit  4>=—\. 

For  3<a<4  as  /3  approaches  positive  infinity  there  are  four  real  roots,  in  the  sheets  AT,, 
Xi,  Pj,  Mj.  Of  these  X2  and  Af,  have  the  limit  <t=i,  the  others  being  infinite.  The  three 
pairs  of  complex  roots  have  an  infinite  limit. 

For  tt<  o  as  y8  approaches  positive  infinity  six  roots  are  real  and  two  of  these,  02  and  ©2,. 
are  finite. 

For  P  approaching  negative  infinity  the  number  and  ordering  of  real  and  complex  roots  is 
essentially  different,  /3=  00  being  a  discriminantal  line.  For  this  approach  and  a>4  two  roots 
are  real  instead  of  ten.     For  a<4  but  positive  four  roots  are  real  and  the  finite  pair  are  con- 


40  THE  PROBLEM   OF   THE  ANGLE-BISECTORS 

tinuously  connected  X2  to  W2  and  M^  to  XXX2.  For  negative  a  only  two  roots  instead  of  six 
are  real  and  these  are  the  finite  pair  00O2  (continuous  with  oo^)  and  ©^  (continuous  with  o^). 
The  discontinuity  at  a  =  4,  ^8=  00  is  essential,  the  original  equation  having  its  last  three  coeffi- 
cients indeterminate  for  these  values. 

The  region  of  ten  real  roots  terminates  at  the  cusp  where  (2)=  (3)=  (4),  (s)=(6)=(7), 
(8)=  (9)=  (10).  A  positive  circuit  of  the  cusp  permutes  the  roots  by  the  substitution  (243) 
(567)  (8,  10,  9). 

On  account  of  the  essentially  incomplete  and  non-analytic  character  of  the  real  field  a  thor- 
oughgoing application  of  the  devices  of  a  Riemann  surface  is  of  course  impossible,  nevertheless, 
as  a  means  of  presenting  the  complicated  state  of  facts  in  a  condensed  form,  it  seems  best  to 
make  a  tentative  use  of  them. 

Drawing  a  barrier  along  <i  =  s  from  /i=2j  to  ;8=  co  and  marking  the  above  substitution 
on  it,  we  name  the  ten  sheets  in  correspondence  with  the  ten  real  roots  in  the  cusp  region : 

ABCDEFGHJK 

127843s        10       96 

This  is  consistent  with  the  barrier  and  gives  the  plan  for  the  real  roots  shown  in  Fig.  10. 

The  complex  roots  need  further  inquiry.  We  conceive  as  many  sheets  as  needed  laid 
over  the  regions  in  question  and  marked  with  the  proper  complex  values.  Z)(a,  j8)  =  o  is  a 
locus  of  points  such  that  three  pairs  of  roots  are  equal.  For  the  real  region  and  also  up  to  the 
four-point  on  the  lower  parabolic  branch  these  have  been  determined.  For  the  rest  of  the  curve 
there  are  two  equal  pairs  of  complex  roots.  [For  k  =  l=i  D  is  rational  in  m  and  the  roots 
corresponding  to  A  are  those  of  a  squared  cubic  factor,  whose  discriminant  vanishes  at  the 
four-points  (one  real)  and  for  m  =  o  which  corresponds  to  1  =  4,  /8=  00  .]     (See  §17.) 

It  is  proper  so  to  name  the  complex  roots  that  after  a  real  pair  become  complex  and  of 
course  conjugate  the  pairing  should  remairt  unchanged  except  by  discriminantal  points. 

The  part  of  D2  where  two  complex  pairs  are  equal  will  effef t  then  an  interchange  of  part- 
ners among  the  pairs,  or  rather  the  monodromie  may  be  so  ordered  as  to  effect  this. 

In  crossing  j8=o  ten  roots  become  zero.  In  all  cases,  however,  four  are  real  and  merely 
have  their  order  of  magnitude  reversed.  The  three  complex  conjugate  pairs  have  the  order 
of  the  pairing  inverted,  the  root  with  positive  imaginary  part  and  that  with  negative  becoming 
interchanged  in  each  pair. 

In  crossing  a  =  o  six  roots  become  infinite,  and  form  a  cycle.  If  we  choose  a  pair  to  be 
real  they  must  not  be  conjugate  but  opposite  in  the  cycle.  This  effects  a  re-pairing  discussed 
in  the  monodromie  for  ^=27  (§14).  Revising  the  numbers  in  respect  of  the  barrier  (3,  4) 
(5,  6)  (9,  10)  leave  the  real  axis  as  conjugates  and  for  a  path  with  fi  constant  crossing  was 
proved  impossible.     Hence  if  (9,  4)  be  the  real  pair  the  hexagon  is 


/         \, 


\/ 

6 

and  (3,  6)  (5,  10)  are  paired  after  crossing. 


THE    TRANSFORMATIONS  4 1 

This  is  for  positive  /3.     A  similar  re-pairing  occurs  for  negative  ^. 

The  phenomenon  is  unlike  anything  occurring  on  a  Riemann  surface  for  an  analytic  function 
of  a  complex  variable.  There  crossing  a  branch  cut  implies  completion  of  a  circuit  round  a 
discriminantal  point:   this  passage  is  only  a  half-circuit  in  the  a  complex  plane. 

The  cycles  given  for  a=  oo  (i,  2,  5,  8,  4,  3,  g)  (6,  10,  7)  to  be  interpreted  as  cyclic  substi- 
tutions invert  the  order  of  pairs  but  keep  the  pairing  of  complex  roots. 

For  fi=  00  to  connect  the  aproaches  with  the  two  signs  the  following  system  of  interchanges 
is  needed: 

0<a<3    (l)  =  (2),    (3)  =  (4), 

(5,  6)  {f,  10)  change  the  order  in  the  pairs, 
(7)  (8)  change  the  relative  order  of  magnitude. 

That  is  a  half-circuit  affecting  pairs  as  on  y8=o. 

3<o<4  in  addition  to  the  above  the  barrier  is  to  be  extended  with  (2,  3,  4)  (7,  6,  5) 
(8,9,10). 

o>4  and  to  end  of  the  upper  branch  ol  D,  (i)=  (2),  (6)=  (5),  (10)  =(9),  (7)  =  (8).«' 

Fora=oo,a'>-  up  to  the  end  of  J's  parabolic  branch  (3)=  (4),  (7)  =  (8),  (i)=(2),  and 
4 

also  the  half-circuit. 

For  the  last  two  fegions  and  also  on  the  rest  of  the  line  /8=  00  ,  a  >o  a  barrier  is  needed  with 
the  substitution  (2,6)  [or  (i,  5)].  For  negative  a  the  order  of  magnitude  is  reversed  and  a 
barrier  (i,  7)  [or  (2,  9)]  is  to  be  applied.  This  barrier  and  the  (2,  6)  barrier  for  a>o  is  needed 
in  view  of  the  occurrence  of  that  part  of  D  where  two  complex  roots  become  equal. 

With  this  set  of  conventions  a  consistent  plan  of  the  sheets  can  be  drawn.  In  Fig.  11 
the  discriminantal  lines  and  the  barriers  are  shown,  with  the  equalities  and  conventional  changes 
in  brackets  [  ]  and  the  pairing  of  complex  roots  in  parentheses  (),  the  first  of  the  pair  being 
the  root  with  positive  imaginary  part.  The  six-cycles  at  tt  =  o  are  symbolically  indicated  by 
hexagons. 

Taken  in  connection  with  Fig.  10  for  the  real  roots  and  the  identification  in  p.  40,  it 
is  to  be  considered  as  a  condensed  expression  of  the  various  connections  between  the  roots  of 
the  equation  for  the  field  of  real  a,  y8. 

Fig.  10  for  the  real  roots  may  serve  the  purposes  of  a  model  of  the  surface  /^(ct,  a,  /3)  =  o 
as  far  as  the  order  of  magnitude  and  number  of  the  real  roots  for  the  various  values  of  (a,  /3) 
is  concerned. 

There  are,  however,  many  questions  which  are  proper  to  ask  concerning  the  connections 
of  the  real  roots  which  it  does  not  answer  or  answers  only  with  difficulty.  For  this  reason  a 
model  is  in  order  to  complete  the  concise  expression  of  the  facts  (§19). 

The  representation  of  the  facts  discovered  in  the  field  of  complex  a,  yS  is  of  course  out  of 
the  question.  For  instance  there  are  two  complex  four-points,  at  which  the  loci  T=o  andZ)=o 
intersect.  These  loci  are,  however,  continua  of  two  dimensions  existing  in  a  space  of  four  dimen- 
sions and  their  intersection  is  a  point  merely. 

To  say  that  this  field  consists  of  the  totality  of  point  pairs  of  two  Neumann  spheres  though 
a  useful  device  for  the  presentation  of  certain  general  arguments  is  not  of  course  a  representa- 
tion which  enables  special  facts  like  the  one  mentioned  to  be  concisely  recorded. 


42 


THE    PROBLEM   OF   THE   ANGLE-BISECTORS 


ot=«o   Cvjdei    LIA,  5,6.'^.'5,9]Lb,10,7J 


FINITE   MULTIPLE   POINTS  .  43 

Infinite  points  of  multiplicity  greater  than  two  occur  at : 

Thepoint£  ;  a  :  b  :  c  ::  i  :  —  : ,  3'  =  z=  — 4, 3<^— 4t  =  o<^=oo    0=4  j8=oo  .    Here 

13         13 
the  sheets  (2)  (3)  (8)  (9)  have  a  common  root. 

Thepointff  ;  a  :  b  :  c  ::  i  :  i:  — i,  y  =  z=  —  i,  <^  =  4,  t=oo,  a=  00,  /?=  00  where  the  sheets 
(3)?  (4))  (7').  (8)  unite.  The  line  a=o  has  six  infinite  roots.  For  the  approach  a>o,  /3>o  all 
six  are  complex,  and  so  for  a<o,  ^<o.  If  a  and  /3  have  opposite  signs  two  roots  have  a 
real  approach.     The  triangles  are  complex  in  all  cases. 

The  origin  is  an  indeterminate  in  (a,  /?). 

For  a  =  o,  ^=0,  -  =  o  there  are  ten  zero  roots.     The  triangles  are  indeterminate  as  t  can 

be  assigned  at  will. 

For  T  =  o)'=|,  2=00  and  the  triangle  can  be  approached  by  way  of 

a=o)oH j —  c=—b{o)+i)- 


20)+ I  20)+ I 

where  «>'=  i  and  b  increases  without  limit. 

ForT=  — i,y=^,2;  =  o.    This  is  the  point  F  which  in  (<^,  t)  is  represented  as  the  line  <^—t=o. 

No  other  triangle  in  the  infinite  set  is  real  except  F  (a  =  b  =  ^  c  =  o).  The  sheets  involved 
are  (3),  (4),  (5),  (7),  (8),  (9),  (10),  and  (9)  is  only  reached  with  t=  co  . 

The  other  approaches  to  the  origin  give  in  some  cases  finite  values  for  o-  but  all  the  triangles 
are  complex,  as  none  of  the  sheets  involved  fall  inside  Z>,  in  the  y,  z  plane. 

/3=o  a^ro  has  ten  zero  roots  independent  of  a,  but  these  can  only  be  approached  in  the 
k,  I,  m  plane  with  complex  values  and  give  of  course  complex  triangles. 

12.      FINITE   MULTIPLE    POINTS 

To  determine  all  the  finite  multiple  points  (other  than  double  points)  a  start  is  made  from 
the  intersections  of  D^  and  T. 

The  corresponding  values  of  <t>  are  given  by 

2<^»— <^+i  =0  for  the  triple  points,  and 
S4<^— S7<)l)'4-24<^— 4  =  0  for  the  four-points. 

The  approximate  values  for  the  real  foiu--point  are: 

<i>=     .4144425  •  •  T=    529566  .  .  . 

a  =  -.99678  .    .  ^=5.3542   .    . 

To  find  the  other  multiple  points  we  write  F(<r,  o,  j8)  =  0  in  the  form 

d.     ,  aF       '     e,     ,  ^F  e,  .^  . 

a  =  -— and  7i-  =  o  as  a  =  — and  7r~  =  o  as  a  =  — i  (60) 

l/'l  OCT  xf>2  "<'■  Vi 

Eliminating  a  in  turn  between  each  pair  of  equations  we  obtain  expressions  (01),  (02),  (12) 
which  must  vanish  for  points  of  multiplicity  three  or  higher. 

If  we  then  write  P=—  and  revert  to  the  (</>,  r)  plane  by  writing  o-  =  <^+i,  (01),  (02),  (12) 

are  of  order  5  in  <^  and  4  in  t. 

(01)  represents  the  discriminant  and  so  T  and  D,  in  their  (<^,t)  form  must  be  factors. 
These  forms  are  given  in  equations  (53)  and  (54). 

As  a  fact  (01)  has  no  other  factors. 


44 


THE   PROBLEM    OF   THE   ANGLE-BISECTORS 


The  form  (12)  is: 


,ps 

^4 

03 

0' 

01 

00 

t4 

—  1920 

—  240 

t3 

3840 

1488 

r' 

—  2712 

-2105 

164 

-77 

T' 

792 

948 

60 

-18 

18 

T" 

-72 

-13s 

-54 

8 

—  2 

—  I 

(61) 


and  the  form  (02)  is: 


05 

04 

03 

0" 

01 

00 

t4 

624 

-48 

t3 

-1584 

112 

-32 

T» 

1377 

—   I 

-   3 

16 

T' 

—  462 

—  102 

36 

-  18 

T" 

45 

39 

-   6 

I 

I 

(62) 


Di  can  be  written 

i6t'' =  32T^</i— i9T<^^+3<^3-|-<^2. 

Reducing  (12)  and  (02)  to  quadratics  in  t  by  means  of  D^  we  have  from  (12)  the  form  I: 

0S  04  03  0J  01  00 


T' 

-432 

196 

164 

-77 

T' 

432 

-414 

45 

-  18 

18 

TO 

-72 

54 

9 

8 

—  2 

—  I 

and  from  (02)  the  form  II: 


05 

04 

03 

0» 

0. 

00 

T' 

-36 

88 

-67 

16 

T' 

54 

-91 

71 

-18 

' 

TO 

-18 

21 

—  II 

—  I 

I 

(63) 


(64) 


EHminating  T  we  have  a  polynomial  in  <^  of  order  16. 

Of  this  the  factors 

(S4<^-S7<^'+24<^-4),  (2<^^-<^+i),  (2.^^+13^+2),  (<^- 2) 
are  known,  the  second  quadratic  and  the  last  factor  entering  as  triple-point  factors  in  the  "equi- 
lateral" case.     24>—  I  appears  as  a  thrice  repeated  factor  and  the  cubic  factor  is  repeated.    The 
residue  is  g<^^+<^— i. 

These  must  include  all  triple,  quadruple,  etc.,  points  on  D,. 

In  the  elimination  between  (12)  and  (02)  which  have  the  form 

common  solutions  of  ^2  =  0  and  i/'3  =  o  enter  which  need  not  satisfy  (01): 

"    -Ac    ^Z 

There  are  five  such  common  solutions  finite  both  ways  and  these  give  rise  to  the  factors 
(2<^— 1)3  {q<l>^-\-<\>  —  i)  after  the  reduction  by  ZJ^.  The  points  they  denote  on  Z)j  are  not  in  fact 
more  than  ordinary  points  on  D^,  that  is  threefold  double  points. 


a  =  ~=—  and  a  = 

"A.       "As 


.(6^^-3'/'+i)  • 


(6<^-i) 


in  (12)  and  obtain  a  polynomial 


For  multiple  points  on  r  =  o  we  write  t  = 

of  order  9  in  <^,  which  has  the  factors 

(54<^^-57<^'+24<A-4)  (2.^"-<^+i)  (12^^- 16<^+7)  (<^-i)  (6^+1) 


THE  DISCRIMINANT 


45 


Similar  operations  on  (02)  give  as  factors  the  same  cubic  and  quadratic  and  a  zero  and 
infinite  factor.  These  and  the  two  linear  factors  of  the  first  set  not  being  common  are  extraneities 
and  the  only  new  points  are  given  by  i2<^'— 16<^+7— o  and  T  =  o.  These  are  triple  points  and 
not  on  D3. 

These  last  points  are  singular  points  on  r  =  o;  being  complex  the  usual  classes  are  without 
significance. 

13.       THE   DISCRIMINANT 

Since  T  =  o  and  D  =  o  are  discriminantal  loci  T  and  D  are  factors  of  the  discriminant  of 
F{<^,a,^)=o.  The  order  of  r  in  u  is  4,  in  ;8,  6.  The  order  of  D  in  a  is  3,  in  y3,  2.  This  is  seen 
directly  in  the  case  of  D  (15)  and  from  the  parametric  form  and  the  elimination  rule  for  T  (57). 
To  obtain  a  closer  view  the  equation  may  be  transformed  by  writing, 


*ZS,  /3  = 


=  4^^^^=<f. 


The  result  is 


5'°— iofo*+i46j'+336V— 94J^5s+(6i6=— 3663  — J'<f)i'"+ 
(i56^»3+6^(/)j3-f-(_2oo63+463(i)j='+(8oJ3-8W)5+4W  =  o 


(65) 


The  order  of  the  coeflScients  in  the  first  derivative  with  respect  to  5  and  a  homogeneity 
factor,  is  in  respect  of  d  given  by  the  rows: 

(o)  (I)  (i)  (3)  (4)  (S)  (6)  (7)  C8)  (g) 


From  this  the  order  of  the  elements  in  Bezout's  form  can  be  readily  obtained  and  inserted 
jn  the  determinant  form, 


The  maximum  order  is  13  occurring  in  the  secondary  diagonal  of  the  first  s-square  and  the 
complete  last  4-square.  A  short  calculation  shows  that  this  order  actually  occurs.  The  factor 
d  cannot  divide  the  result  as  <i  =  o  is  not  discriminantal  for  64=0. 

The  order  in  d  is  the  order  of  a  as  far  as  it  depends  on  the  factors  T  and  D;  a  =  o  correspond- 
ing to  d=<x>  being  represented  by  the  defect  of  this  order  from  18  the  order  of  the  general  case. 
As  a  =  00  is  not  a  discriminantal  locus  as  must  occur  as  a  factor  of  the  discriminant  of  F(a;  a,  p) 
=  0.  The  orders  of  am  T,  D  being  4,  3  we  have  4tM+3«=  13  where  m,  n  are  the  exponents  of 
T,  D  in  the  discriminant.     The  only  solution  is  w=  i,  «  =  3. 

As  to  the  powers  of  j8  a  count  of  order  in  the  Bezout  form  for  F{<j,  a,  /3)  gives  44  as  the 
maximum  and  of  these  30  can  be  divided  from  the  rows  and  columns.  Since  ^  enters  T.D^  to 
the  1 2th  order  the  exponent  of  /8  is  either  30,  3 1 ,  or  3  2 .  The  fact  that  f or  ;8  =  o  no  complex  roots 
become  real  or  vice  versa  bars  31. 


46  THE  PROBLEM   OF   THE   ANGLE-BISECTORS 

To  decide  between  30  and  32  it  is  necessary  to  examine  the  monodromie  cycles  for  /3=o. 
In  the  neighborhood  of  /3  =  o  the  equation  to  a  numerical  factor  is  properly  approximated  by 

[a3+J/3]  [<Ti+t,P]  l<J'+hl3]  [<T+w/3]  =  0 
20ft 

where  m= , r  and  t^,  t^  are  the  roots  of  4t'  —  gt+4  =  o. 

(a -4) 

Of  the  45  squares  of  differences  of  the  roots  9  are  differences  of  roots  belonging  to  the  same 
3-cycle.  These  differences  each  vanish  with  /8  and  to  the  order  5.  The  product  of  their  squares 
vanishes  to  order  6.  There  are  27  differences  of  roots  from  different  3-cycles  and  these  together 
contribute  Z?"'.  The  9  differences  of  roots  of  a  3-cycle  and  the  odd  root  are  also  of  order  §  and 
contribute  )8*. 

The  totar order  to  which  the  discriminant  vanishes  with  j8  is  then  30.  It  is  clear  that  this 
method  of  determining  the  exponent  of  a  discriminantal  factor  is  general,  and  depends  for  its 
effectiveness  only  on  the  determination  of  the  cycles.  The  only  converse  to  the  theorem  is 
that  odd  exponents  and  odd  substitutions  of  the  roots  are  correlated. 

The  complete  discriminant  has  the  form 

The  determination  of  the  numerical  factor  N  has  proved  impracticable. 

14.   THE  MONODROMIE  GROUP  OF  THE  EQUATION  F(<T,  a,  ^)  =  0 

To  simplify  the  numerical  work  write  (r  =  2s,  o.  =  2,a,  li  =  ib. 
The  equation  becomes 

1605'°— i2oa65'-|-56a65'+297a6^5^— 282c6^55-j-6ifl&V—  2']oahis^-\-j,()his^-\-T,()cah^s^     \   ,     , 
—  1 263^^—1  SoaZ)''i"+8iaZ)V— io86V+2oa635  — 54a6''5 -1-726''^ -fgai)''— 126'' =  0     j 

We  begin  by  placing  b=i 

F(5,a,  i)=a(i65'°—i205*+56.J^+2975*— 28255  — 2095"-}-  | 

36053  — 695^  — 34.y+9)-fi2(3s''— 53  — 9s^-|-65—i)  =  o  ( 

For  a  =  I ,  i.e.,  the  cusp  on  D{a,  /?)  =  o  the  values  of  s  are  given  by 

25-3=0,    (5-l)3=D,    (253-1-35-1)3  =  0  (§S). 

For  a=  —  I  also  on  D  they  are  given  by 

(453-45^  —  55-1-2)^  =  0,  53—25-1-3  =  0,  S-\-2=0. 

Reference  should  be  made  to  the  graph.  Fig.  12. 

To  discuss  the  connections  of  the  roots  at  5  =  1,  a—i,  b  =  i  we  write  5=i-t-^',  a  =  i+a', 
b  =  i+b'  and  retain  terms  of  the  first  three  orders  as  this  is  a  triple  point. 
We  have  as  a  proper  approximation  to  the  curve,  dropping  the  accents, 

6a—2b+2Sab+i4as+bs—iib^+64abs+4S^s'—2ob's  —  i6si+43'^b'—i6bi  =  o. 

If  the  origin  is  approached  along  the  plane 

6a  — 26  =  0  the  (a,  s)  projection  is 
1705- 99a^-|-i2a^5-|-ii7a5'— 1653  =  0. 


MONODROMIE  GROUP   OF  F   {o;  a,  ^)=o  47 

The  approximations  at  the  origin  are  given  by  Newton's  parallelogram  as 
1705— 99fl"  =  o  and  —1653+17^5  =  0. 

The  first  branch  gives  a  stationary  root  while  the  second  has  a  parabolic  factor  correspond- 
ing to  a  possible  interchange  of  two  roots. 

In  fact  i6i^  =  i7a  and  if  a  the  parameter  moves  round  the  origin  in  its  own  complex  plane 
the  complex  values  of  s  each  move  round  a  half-circuit  in  the  s  complex  plane  and  are  so  inter- 
changed. 

We  have  then  for  this  point  and  this  approach  an  interchange  of  a  pair  of  roots  entering  as 
part  of  (a  cycle  of)  a  substitution  of  the  roots  which  is  an  element  of  the  monodromie  group. 


At  the  same  point  but  by  an  approach  on  a  =  kb,  k+l  gives  8^^=  (3^— 1)6  and  a  circuit  in 
the  b  complex  plane  interchanges  three  roots  cyclically  and  these  three  roots  must  include  the 
former  two,  by  the  principle  of  the  continuity  of  the  roots,  and  the  fact  that  the  singular  points 
are  necessarily  distinct. 

The  pair  of  roots  is  real  for  real  a's  and  of  the  set  of  three  one  is  real. 

The  same  values  of  the  parameters  a  and  b  give  two  other  triple  roots,  one  set  being  at  s 

— - — ' — —  .    A  similar  treatment  gives  in  this  case  also  a  cycle  of  three  for  the  general  approach 
4 

and  a  transposition  for  approach  in  the  tangent  plane.     So  also  for  the  conjugate  set. 

Since  the  tangent  planes  are  distinct  in  the  three  cases  we  have  as  elements  of  the  group  for 
a  general  approach  a  substitution  of  the  form  (234)  (567)  (89,  10),  while  for  the  tangent 
plane  approaches  we  have  the  elements  (23)  (567)  (89,  10);  (234)  (56)  (89,  10);  (234) 
(567)  (89)  respectively.  The  roots  entering  the  transpositions  may  be  any  pair  of  the  corre- 
sponding set  of  three. 

After  the  cycles  at  a=  i,  b=i  the  monodromie  path  is  taken  along  b=i.  To  determine 
whether  any  multiple  points  are  encountered  we  eliminate  a  between  F(s,  a,  i)  and  its  derivative 


(l) 

5=1 

•57 

(2) 

S  = 

•347 

(s) 

S  = 

■333 

(8) 

s=  - 

-1.9 

48  THE  PROBLEM  OF  THE  ANGLE-BISECTORS 

with  respect  to  5.  The  resultant  which  is  of  order  13  in  s  must  include  all  finite  multiple  points 
on  6=1.  Of  these  5=  i  and  the  roots  of  25^+35— 1  =  0  are  known,  and  in  fact  they  are  repeated. 
The  remaining  septimic  has  as  a  factor  45^—45^—  55-I-  2,  which  gives  the  values  at  a=  —  J  on  Z). 
The  residue  is 

185"— 753— 5452-1-455—12  =  0. 

This  quartic  has  two  complex  roots  and  two  real  whose  approximate  values  are  5=1. 4598  .  . 
and  5  =  —  1.924  .  .  which  correspond  to  a  =1.47  .  and  a  =  —  1.27  .  respectively.  (These 
points  are  on  the  upper  and  lower  branches  of  r=  o.) 

There  being  no  multiple  points  between  a=  i  and  a  =  o  we  note  that  at  a=o  the  four  real 
roots  are 

which  at  a=  I  had  the  value       i .  5 
which  at  a  =  I  had  the  value      i 
which  at  a=  I  had  the  value         .  28 
which  at  a=  I  had  the  value  —  i .  78 

Since  the  equation  is  of  the  first  order  in  a  roots  can  only  cross  at  double  points.    This 
applies  to  complex  roots  as  well  as  real  for  no  different  a's  can  have  the  same  value  of  5. 
At  a  =  o  six  roots  which  were  complex  at  a=  i  become  infinite. 
The  finite  roots  for  a= o  being  given  by 

(35-i)(53-[35-i]6)  =  o 

the  root  marked  (5)  has  a  fixed  value  independent  of  b. 

The  cubic  factor  has  equal  roots  at  b=\  when  (i)  and  (2)  become  equal  and  as  the  origin 
is  approached  with  a  constantly  zero  the  three  roots  (i)  (2)  (8)  can  be  made  to  take  part  in  a 
cycle  by  means  of  a  circuit  of  b  round  its  own  origin  in  the  b  complex  plane. 

To  determine  the  configuration  of  the  roots  at  a  =  o,  s=  00  we  notice  that  at  a=  i+c',  a' 
small  the  roots  (3)  (4)  are  conjugate  complex  roots  of  8x^—30'  =  o  and  as  in  the  monodromie  path 
from  a=i  to  a  =  o  we  leave  the  point  i  by  decreasing  a,  a'  is  negative.  The  real  root  is  then 
less  than  i,  and  the  complex  roots  have  a  real  part  greater  than  i.  We  may  choose  (3)  as  the 
root  with  positive  imaginary  part. 

In  the  same  way  it  may  be  shown  that  the  roots  (5)  (6)  (7)  have  real  parts  less  than  i  and 
(6)  may  be  taken  as  the  complex  root  with  positive  imaginary  part  and  (5)  as  the  real  root. 

For  (8)  (9)  (10)  the  real  part  is  negative  and  we  take  (8)  as  the  real  root  (9)  as  the  complex 
root  with  positive  imaginary  part. 

As  a  leaves  the  point  i  and  decreases  the  paths  of  the  six  complex  roots  in  the  s  complex 
plane  start  in  the  order  indicated  in  the  diagram  (Fig.  13). 


"1 

Now  F{s,  a,  i)  contains  a  in  the  first  order  only,  so  no  different  a's  can  have  the  same  value 
of  5,  and  if  the  paths  cross  it  must  be  at  a  multiple  point.    There  are  however  no  multiple  points 


EQUATION   FOR   THE   SIDES  49 

for  6  =  I  and  a  between  i  and  o.    The  six  complex  roots  then  reach  the  infinite  point  of  the  s 
plane  in  the  reversed  cyclic  order  and  since  by  writing  5  =  -  the  proper  approximation  is 

40+9/''=  o 

they  there  enter  into  a  cycle  (3,  4,  7,  10,  g,  6). 

If  we  pass  along  6=  I  for  a  >i  we  reach  a  double  point  at  a=  1 .47   .    .   where  s=i  .4508  .  . 
is  the  double  value. 

Since  at  a=  1  the  root  (i)  has  the  value  i .  5  and  the  root  (2)  the  value  i  and  no  double 
points  occur  in  the  interval  we  add  the  transposition  (12)  to  the  elements  of  the  group. 

At  a=  —  1 .  27  .  .  is  another  double  point.  To  identify  the  roots  here  we  suppose  that 
the  six-cycle  at  a  =  o  is  so  passed  as  to  leave  (3)  and  (10),  which  are  opposite  in  the  cycle,  in  the 
real  positions  —  00  and  +  00  respectively,  when  the  added  element  will  be  (3,  8). 

At  a=—i  there  are  three  pairs  of  equal  roots  of  which  (25)  and  (i,  10)  are  two.  The 
third  pair  must  be  opposite  in  the  six-cycle  at  a  =  o  and  is  then  either  (94)  or  (67).  It  is  in 
fact  immaterial  which  is  taken. 

As  elements  of  the  monodromie  group  we  have: 


A 

(12) 

ata=   1.47 

5, 

(23)  (567)  (89>  10) 

at  a=   I 

B, 

(234)  (56)  (89,  10) 

B^ 

(234)  (567)  (89) 

C 

(234)  (567)  (89,  10) 

at  a=   I 

D 

(347.  io>  96) 

at  a=  0 

E 

(i,  10)  (25)  (49) 

ata=-| 

F 

(38) 

at  a=  —  1.2; 

The  transposition  (12)  exists.     By  transforming  it  by  Bi  we  add  (13). 
Transforming  this  by  C  (14)  is  reached. 
Transforming  (13)  by  F  we  reach  (18). 
Transforming  (18)  by  B^  we  reach  (19)  and  (i,  10). 
Multiply  £  by  (i,  10)  and  obtain  (25)  (49)-. 
Transform  (12)  by  this  and  obtain  (15). 
Transform  (15)  by  5,  and  obtain  (16)  and  (17). 

We  have  now  every  (i,  n)  from  which  every  single  transposition  and  the  symmetric  group 
can  be  obtained. 

By  Jordan's  theorem  the  group  of  the  equation  is  the  symmetric  group. 

15.      THE   EQUATION   FOR    THE    SIDES 

To  obtain  an  equation  for  the  value  of  the  side  of  a  triangle  with  given  internal  angle- 
bisectors  we  write  a-\-b-\-c=  1  and  use  ratios.     We  have 

„_l ^    i-2a      b{b-iY 
~k~a{a-iY      I- 2b  , 

m      I  — 2a      c{c—iy 


5=v= 


k     a{a—iy      I  —  2C 


50  THE   PROBLEM    OF    THE   ANGLE-BISECTORS 

Using  N=—, r-  as  an  abbreviation  we  obtain 

a{a—iy 

{l-2b) 

Write  now  bc  =  p  and  note  that  b+c=i  —  a  so  that  b,  c  are  the  roots  of 

u'—{i  —  a)u+p  =  o  (69) 

Expressing  the  symmetric  function  of  b,  c  in  terms  of  the  coefficients  of  this  equation 

R+S  =  N  .  [-a-+a'+P(-2«-+3a-i)+4P-]  ^    ^ 

4P+2fl— I  ' 

Similarly 

4P+2a— I  ^' 

We  now  vfTite  RS=p,  R+S=q,  and  p  =  m—a 

N'ni'im—a)  _7V[4m'— w(2a'+5a+i)+a+i)'a] 

4W— (2a+i)  4OT— (20+1) 

If  now  -=  r  we  have  a  cubic  and  quadratic  for  m. 


-m 


Nm^-\-m^{—  aN  —  i^r)-\-mr{2a^-\-  ^a-\-i)  —  r{a-\-iya  =  o 
4Nm'—in{N[2a'+sa-\-^]-\-Aq)+N[a{a+iy->rq{2a-\-i)]  =  o 


in) 


Using  the  indicated  end-term  multipHers  whose  determinant  is  4  no  extraneities  are  intro- 
duced and  a  second  quadratic  results. 
If  this  is 

A'm^-\-B'm-\-C'  =  o  (74) 

and  the  previous  quadratic  is 

Am^-\-Bm-\rC  =  o 

the  second  order  determinants  which  enter  the  eliminant  are 

AC'-A'C=Aq'{2a+i)+i()qr{2a+i)-Nq{&a^-\-i2a'-\-']a+i) 

-m{2a^+Sa'+Sa'+^a'+a)  (75) 

'       AB'-A'B=i(yq'-(i/^qr-\-j^NqUa'+Aa+-i)-\-N'{Aa'-\-?'ai+a'+2a+i)  (76) 

BC'-B'C=q'(,2a-\-iy+2Nql2a+i){a+iya-\-^qr{-4a^-T,a-i)+N^{a+xYa^    (77) 

Collecting  the  coefficients  of  the  powers  and  products  of  N,  q,  r  the  eliminant  takes  the  form 

Q=  —  ibq^ria) 

-q^'N{2a-\-\y{2a-\) 

^-bAqVia) 

-f49^'-iV(2a+i)^(a-i)  . 

-9W^(2a+i)(6a3-j-5a2-a-2)a  '    ^'  ' 

-qm{a-\-iy{6a>+a'—2a-i)a' 

+griV^(20-t-i)(6a3-f-7a^+4a— i)(a— i) 

-N^{a+iy{a-i)a\ 

and  writing  for  N  its  value  in  terms  of  a,  multiplying  throughout  by  (a—  i)'  a^  and  putting  qr  =  p 
we  have  the  eliminant  as  an  equation  of  the  tenth  order  in  a,  containing  two  parameters  p,  q 
among  its  coefficients.    Arranged  in  powers  and  products  of  ^,  9  it  is : 


EQUATION   FOR   THE   SIDES  SI 

F(a  :  p,q)  =  q'{2a+iy(2a—iy(a—iya 
—  i6pq'(a—  i^a^ 
+  64/>^(a-i)'a3 

-4pq(2a+iyi2a-i){a-i)''a  ,  , 

-q'i2a+i){6ai+Sa'-a-2)i2a-iy{a-iya  '    ^'^' 

+9(6a3+a'+2a-  i)(2a-  i)^(a+  i)3(a-  i)a 
+/>(2a+  i)(2a-  i)^(6a<+a3- 30^-  sa+  i)(a-  i)^ 
-{a+\y{2a-\ya^ . 

The  result  may  be  checked  by  substituting  />=  i,  9=  2,  i.e.,  k  =  l=m  when  it  reduces  to  give 

1     (3=^V  17)       (i*Jl£7)       (l^Jiil) 
4         '  8         '  8 

which  agrees  with  the  previous  result  for  this  case. 

To  obtain  6  as  a  rational  function  of  a  the  order  of  eltmination  must  be  changed.      With 

this  order  m  and  hence  be  is  obtained  as  a  rational  function  of  a. 

R  9 

If  we  write  ^=/;  T;=i?>  we  have  two  cubics  in  b: 

N        N 

b}-2b'+bii+2f)-f=o, 

b>+b'{:ia-  i)  +  6(3a^—  2a+  2g)c+  {ai—a'+  2ag—  g)  =  o, 

by  carrying  the  elimination  to  the  penultimate  step  by  the  end-term  method  we  have 

6[-2/'-|-2g^(2a-i)-4/g(a-i)+g(8a3+3a^-2a-i)-2/(a+i)^a+(a'-i)(3tf+i)a"] 
+  [g"(2a-i)'+2/g(-6a'-i)+/'+^(2a-i)(2a3— 2a'-3a-i)+/(-6c''-3a3+3a'+2a) 
(a^—  i)(a'—  2a—  i)a"]  =  o. 

In  this  expression 

IV  (a-i)'g~|  mV  {a-iya  "1 

■^~^L(-2a+i)J  ■  ^~'^  L(-2fl  +  i)J 

making  these  substitutions  and  dividing  by  ,    ,^  we  have  unless  a=  i,  5,  or  o 

b[—  2t'{a—  lya— 4lm{a—  i)''a+2W^(a—  i)J(2a—  i)a-|-2/^(a-|-  i)^(a—  i)(2a—  i)a— 

mk{a—i)  (2a— i)  {&a^+sa'—2a—i)+k^{a+i)  (30+1)  (2a— i)^a]  \  ,„  •. 

+  [l'(a-iya+2lm(a-iy(-6a'-i)a+lk{a-i){2a-i){6ai+Sa'-Sa-2)a+k^{a+i)         ( 
(a'—  2a—  i)(2a—  iya-\-m'{a—  i)-'(2a—  iya—mk{2a—  i)^(a—  i){2a^—  2a^— 3a—  i)]  =  o  ' 

In  obtaining  approximate  solutions  substitution  in  this  expression  is  more  laborious 
than  the  solution  of  a  cubic.  Hence  on  this  ground  the  solution  by  the  o-  chain  (11)  and  (3) 
is  preferable.  In  practice  however  the  method  of  trial  on  the  equations  (22)  is  more  expedi- 
tious than  either. 

Theoretically  we  note  that  the  sides  involve  merely  an  irrationality  of  the  tenth  degree 
and  the  cubic  which  occurs  in  the  o-  chain  is  not  necessary  but  convenient.  The  cubic  irra- 
tionality is  then  not  accessory  in  the  technical  sense. 

p{a  :  p,  q)  is  of  course  unsymmetric  m  {k,  I,  m)  and  as  />  =  ,  ^ ,  q=  — 7 — ,  it  can  be  denoted 
by  F{a,  k  :  l,m)  and  is  associated  with  two  other  equations  by  cyclic  change. 


52  THE   PROBLEM    OF    THE   ANGLE-BISECTORS 

That  these  equations  which  have  the  same  form  are  all  irreducible  can  be  established  thus: 

Suppose  they  reduce  in  {k,  I,  m),  choose  a  corresponding  set  of  factors,  and  from  them  form 
the  equation  whose  roots  are  o-  expressed  as  a  symmetric  function  of  a,  b,  c. 

The  coefficients  of  this  equation  will  be  symmetric  functions  of  k,  I,  m  of  order  zero  and 
hence  rational  in  a,  /8. 

For 

kl-\-lm-\-mk  klm 

and  writing  k+l+m  =  P,  kl+lm+mk  =  Q,  klm  =  R  every  coefficient  when  the  denominators 
are  cleared  will  be  the  sum  of  multiples  of  such  terms  as  R^Q^P^  where  on  account  of  the  homo- 
geneity 3^+2);+^=w. 

On  dividing  such  terms  by  P"  we  obtain  terms  of  the  form 

R^Q^  _  R^Q^ 


Pl-i         Pii+2r\ 


=  l3-(a- 


The  equation  will  be  satisfied  by  the  corresponding  values  of  o-  if  the  equations  in  a,  b,  c 
are  satisfied  and  will  be  of  an  order  less  than  lo.  But  F((t,  a,  yS)  which  will  be  the  result  of 
carrying  out  the  above  process  on  all  the  factors  oi  F  {a,k  :  l,m),  etc.,  does  not  reduce,  which 
contradicts  the  hypothesis  of  these  equations  being  reducible. 

l6.   THE  MONODROMIE  GROUP  OF  THE  EQUATION  FOR  THE  SIDES 

The  form  in  which  the  equation  emerges  from  the  elimination  (79)  renders  the  determina- 
tion of  critical  points  with  the  corresponding  binomial  approximations  and  cycles  of  the  roots 
easy. 

We  deal  first  with  the  values  p='xi ,  q=cc  .     H  p  =  kq  and  A  is  finite  the  common  factors 

of  the  terms  of  highest  order  in  q  are  (a  —  i )%.     For  a=i  =  n,q=-  the  terms  to  be  considered 

are  of  orders  «',  kw^,  k^u,  k^. 

The  first  three  give  2  two-cycles  and  the  last  pair  a  stationary  root  at  a  =  i ,  there  is  a  single 
root  at  a=o  and  four  other  roots  dependent  on  A.  and  in  general  distinct. 

This  approach  to  the  point  p—  00,  9=  <»  gives  then  an  element  of  the  group  of  the  type 
(12)  (34). 

If  we  write  ^p  =  q^  and  then  9=-  we  get  the  same  state  of  things  at  fl=  i  but  at  a  =  o  there 

is  a  two-cycle,  the  other  roots  are  an  odd  one  at  a=  5  and  a  pair  not  forming  a  cycle  at  a=  —  5. 
Since  on  the  Neumann  spheres  for  p,  q  the  points  used  are  as  close  as  we  please  we  may  iden- 
tify the  totality  of  roots  at  a  =  i  in  the  two  cases.  For  a  real  approach  the  two  pairs  of  roots 
at  a=  I  are  as  near  as  we  please  two  pairs  with  equal  values  and  opposite  sign  and  as  this  char- 
acter is  the  same  for  all  finite  values  of  A.  no  discriminantal  branch  is  crossed  and  the  pairing 
may  be  identified  in  the  two  approaches.  By  the  product  of  the  two  substitutions  a  single 
transposition  is  obtained. 

For  the  approach  with  A.=o,  or  by  writing  q=o  and  then  p=~  we  have  for  a—i  =  n  a 

TT 

binomial  approximation  of  the  type  «'  =  5r"  giving  a  seven-cycle,  and  for  a  near  o  a  three-cycle. 
By  the  continuity  of  the  roots  and  the  approximate  equality  of  p,  q  in  the  cases  A.=o  and 


REDUCTION   OF   EQUATION  FOR   O"  IN   CASE   OF   EQUAL   BISECTORS  53 

X  finite  but  small  we  may  identify  the  two  roots  which  enter  the  single  transposition  with  two 
of  those  in  the  three-cycle,  and  conclude  that  the  seven-cycle  contains  the  four  roots  which 
entered  the  pair  of  two-cycles  in  the  other  approaches. 
We  then  have  as  elements  of  the  group 

A  :  (afiySe!:r,)(eaK)  and  B  :  {Oa) 

At  the  point  q  =  \p=o  there  is  a  four-cycle  at  a=  —  i,  a  second  four-cycle  at  a  =  ^,  and  a 
two-cycle  at  a  =  o.     This  gives  the  element 

C  :  {abcd){efgh){ij) 

From  combinations  of  A''  and  B  the  subgroup  G(,{6<jk)  is  generated  and  one  of  these  roots  must 
be  connected  by  C  with  some  root  in  the  set  of  seven  in  A^.  By  using  the  powers  of  A^  as  trans- 
formers every  single  transposition  of  the  ten  roots  can  be  produced  and  hence  the  group  do! 
The  monodromie  group  being  the  symmetric  group  by  Jordan's  theorem  the  algebraic 
group  is  also  the  symmetric  group. 

17.   THE  REDUCTION  OF  THE  EQUATION  FOR  w  IN  THE  CASE  OF  EQUAL  BISECTORS 

In  this  case  two  angle-bisectors  are  equal.     If  K  =  L  and  Ti^='» 

{m+2Y         „     {m+2y 

a= 1 li= . 

Taking  m  as  the  single  parameter  and  substituting  for  a,  /S  and  writing  <T  =  ^[m-\-2)  the 
equation  when  divided  by  (w+2)"°  becomes 

-|- (40w^-|- iS2w-f  156)^^-1- (w?5— 62W— ii6)^^-j- (—w^-f4W-f-36)f-f  (w— 4)  =  o 
Since  in  general  y8=  — ,  o-=  w-f  2,  T=m\s  suggested  as  a  solution.     This  gives  i=  i  which  satis- 

T 

fies  the  equation  and  dividing  out  the  factor  we  have: 

i6m^i^-\-\bm^i^-\-{—2/^m^  —  9>om^)iT^{—2^m^—2i\ni')^^-\-{gm^-\-io%ni'-\-i^2m)i^    \ 

+  (9OT3-|-i4w"-56m)l'>-f  (-w3-38w^-95OT-72)^3+(-»M34-2W^H-57W-f84)^'        (82) 

+  (2w'— sw— 32)^4- (-w-f4)  =  o  / 

This  must  be  of  the  form 

where  the  coefficients  are  polynomials  in  m.     We  see  at  once  that  D=i.     Recalling  the  special 
cases 

a=3,  /3=27,  w=i 

a  =  4,  /3=54,  w=4 
where  the  second  factor  is 

4^^+o^^-iif-|-3 
16^34-0^- 2ol-fo 

respectively,  and  seeing  that  the  polynomials  cannot  be  of  higher  than  the  first  order  in  m  we  try 

4mi'+o^'-(sm+8)i-im-4)  (83) 

which  divides  with  quotient 

[2mii+mi'-  im+s)i+iY  (84) 


54  THE  PROBLEM   OF   THE  ANGLE-BISECTORS 

The  solutions  for  D{a,  li)  =  o  are  then 

i)  <T=2-\-in,  T  =  m  :  which  gives  y=<=^,  z=oo,  and  an  infinite  triangle  whose  sides  are 
[5,  +  00 ,  — "»  +5]  a  solution  for  every  m. 

This  has  been  discussed  in  the  case  of  the  equilateral  triangle. 

2)  The  factor  4m^'—  (^m+8)i—  (m— 4)  gives  three  isosceles  triangles.     Its  discriminant 

being 

28 

-m{2'jm'+gm+52)  (85) 

there  are  three  real  roots  for  positive  m's,  one  real  root  for  negative  m's. 
The  distribution  of  m  over  D(a,l3)  is  as  follows: 

o<m<i  from  a  =  4, /3=  00  to  the  cusp 

i<m  from  the  cusp  to  a=  00  ,  /3=  00 

o>w>  — 5  the  hyperbolic  branch  in  the  second  quadrant 

—  §>w>  — 2  the  branch  in  the  third  quadrant 

—  2>m  the  parabolic  branch  in  the  fourth  quadrant 

The  origin  is  w=  —  2,  the  asymptotic  points  m  =  o,  m=—\. 

For  o<m<  i  the  three  real  roots  of  this  factor  are  those  called  1,  5,  10. 

For  m>i  they  are  i,  7,  8. 

The  root  (i)  gives  an  isosceles  triangle  with  internal  bisectors  equal. 

The  root  (5)  gives  an  impossible  triangle  with  real  bisectors. 

The  root  (10)  gives  a  real  possible  isosceles  triangle  with  equal  external  bisectors  at  the 
base  which  is  the  smallest  side.     The  larger  bisector  is  internal. 

The  root  (7)  gives  an  impossible  triangle,  while  (8)  has  a  real  isosceles  triangle,  the  equal 
bisectors  being  external,  the  smaller  one  internal.  . 

For  negative  m's  if  w>  — |  the  point  representing  the  real  solution  (which  gives  a  complex 
triangle)  falls  in  XXX 2.  In  the  other  cases  the  triangle  is  real  and  impossible,  f or  —  2  <  w  <  —  5 
the  point  falls  on  the  boundary  of  ©2  and  XXX ^,  for  w<  —  2  on  the  boundary  of  ©,  and  XXX y 

3)  In  the  case  of  the  squared  cubic  factor  the  discriminant  is 

—  [gw^+aSW-t-QW-f  216]  (86) 

27 

For  m>o  there  are  always  three  real  roots.     The  cubic  factor  of  the  discriminant  has  one  real 
root  only, 

w.= -4.987  .   .  ,  a=-.997  .   .  ,  /3=5.3S4  .  .   .      (See  §12.) 

For  m<m,  there  are  three  real  solutions,  for  o>w>Wi  only  one.  Along  the  latter 
part  of  D{a,P)  two  pairs  of  complex  roots  become  equal.  The  point  w,  is  the  crossing  point 
of  T,  D2  in  the  <i>,  r  plane  and  a  point  of  tangency  of  T  and  D  in  the  a,  y8  plane:  the  real  four- 
point.     For  o<w<i  the  solutions  fall  in  (2,  3),  (6,  7),  (8,  9). 

The  (2, 3)  solution  is  a  real  possible  triangle  with  one  of  the  equal  smaller  bisectors  internal. 
The  (6,  7)  solution  is  impossible.  The  (8, 9)  solution  like  the  (2,  3)  has  one  of  the  equal  smaller 
bisectors  internal  but  the  smallest  side  is  much  smaller,  the  opposite  angle  being  always  <  20°. 

For  I  <  w  <  00  the  solutions  fall  in  (3,  4),  (5,  6),  (9,  10). 

The  (3,  4)  case  has  the  less  bisector  internal.    The  (5,  6)  triangle  is  impossible. 

The  (9,  10)  case  has  one  of  the  larger  bisectors  internal. 


REDUCTION   OF  EQUATION  FOR   THE   SIDE   IN   THE   CASE   OF  EQUAL   BISECTORS         55 

In  the  case  of  isosceles  triangles  it  is  obviously  unnecessary  to  solve  two  cubics.  The 
cubic  for  the  side  a  is 

4(w  — 4)a^— (9OT  — i6)a^+2(3w— 2)a— »j  =  o 
whose  discriminant 

—  (27OT'+9W+32)  (87) 

27 

is  algebraically  identical  with  the  i  discriminant  involved. 
In  this  case  b  =  a,c=i  —  2a  completes  the  solution. 

[For  K  =  L,a-=b,mM'  =  K^] 

18.      REDUCTION   OF   THE   EQUATION   FOR   THE    SIDE    IN    THE   CASE    OF    EQUAL   BISECTORS 

2W  m' 

If  l=m,  q  =  -j-,  p  =  -^. 

Writing  9  =2/?,  p  =  R' in  the  equation  the  coefficient  of  R''  vanishes  identically  and  (2a— i) 
is  a  factor  of  the  remaining  terms. 
The  residue  is 

8i?3(2a+i)^(a-i)sa^+i?^(2a+i)(2a-i)(a-i)3(-i8ai-i9a3+a^+3a+i)  (   ,gg, 

+  2/2(2a-i)Ka+i)2(a-i)(6a3+a^-2a-i)-(a+i)H2a-i)3a='  =  o  ( 

The  case  of  isosceles  triangles  gives 

SR{a—i)a^—(2a  —  i){a+i)''  =  o  as  a  factor. 

The  other  factor  is   RH2a+i)'{a-i)' -  2R{2a+i)  (20-1)  ia-i)'{a+i)a+ (a+i)' 
(2a—  i)^a^.     This  is,  as  it  should  be,  a  square. 
For  l  =  tn  v>e  have  then  for  a 

i)  a  =  i  giving  the  infinite  triangle  discussed  in  (§  5) 

2)  ;t(a+i)^(2a-i)-8w(a-i)a^  =  o  (89) 

for  the  three  isosceles  triangles,  b  =  c  = . 

3)  Three  triangles  each  a  double  solution  from 

k{a+i)  (2a— i)tt— w(2a+i)  (a— i)"  =  o  (90) 

In  the  last  case  after  finding  a  we  have  a  quadratic  for  b  and  c. 

/For  6+c=i— a  and  since  h  —  nic;  Diib,  c)  =  o 
{b  +  cY-2bcib  +  c)  +  3bc-2ib  +  c)  +  I=0  (40) 

from  which 

a' 
bc=—       ,  -  and  with  6+c=i— a 
2a+i 

a  quadratic  for  b,  c. 

Hence  the  solution  of  a  cubic  and  quadratic  is  sufficient. 

It  will  now  be  shown  that  no  simpler  solution  exists  for  a  general  value  of  -:  . 

If  we  discuss  the  corresponding  factor  for  F(a,  k  :  l,m)  for  k  =  l  the  method  of  elimination 
(§  19)  obviously  fails  to  distinguish  a  and  b  in  any  way  and  we  get  a  sextic  factor  for  the  Dj 
case  k=l,  a^b. 


REDUCTION  OF  EQUATION   FOR   THE   SIDE   IN   THE   CASE   OF   EQUAL   BISECTORS        57 

This  factor  gives  all  the  a's  and  b's  needed  to  make  up  the  three  triangles  involved,  the 
connection  of  the  pairs  being  determined  by 

D,{a,b)  =  ia+by-2ab{a+b)+sab-2{a+b)  +  i=o  (40') 

This  rational  relation  holding  for  three  pairs  of  the  /oots  the  group  of  the  sextic  reduces. 
In  fact  we  have  shown  how  by  solving  a  cubic  for  c  to  find  a  and  b  by  solving  a  quadratic.  If 
the  sextic  is  irreducible,  the  group  is  transitive  and  cannot  further  reduce  than  is  indicated  by 
this  solution. 

The  sextic,  obtained  by  a  process  similar  to  that  which  afforded  the  corresponding  cubic,  is 
OT'[(a-i)3(8a»-4a-i)a]+OT/[(a-i)(-8ai+4a'-4a'+Sa-i)o]+/"(3a^-i)^  (91) 

The  other  factor  is 

w(4a—i)(a—i)'— 4/(20— i)'a  (92) 

the  tenth  root  being  infinite. 

The  irreducibility  of  the  sextic  is  easily  established. 

If  the  roots  are  paired  as  (i,  2)  (3, 4)  (5, 6)  we  may  write  the  rational  relation  Diii,  2)  =  0 
shortly  as  (12)  =0. 

(The  function  (i  2) +  (34)  + (56)  =  0  is  in  G^i  and  distinct  from  its  conjugates.) 

The  group  is  as  in  general  in  the  case  where  a  general  cubic  with  a  parameter  and  a  quadratic 
give  rise  to  a  sextic  on  eHmination  G^g generated  by  the  substitutions  (12) :  (i35)(246)  :  (i3)(24). 

As  the  Galois  resolvent  may  be  taken  the  equation  of  degree  48  rational  in  m  which  has  for 
roots 

a,  — ^,+(0(02  — 62)+«^(a3  — 63)  and  its  conjugates. 

Each  root  may  be  rationally  expressed  in  terms  of  any  one  of  these. 

It  is  interesting  to  note  what  happens  to  the  rational  expression  of  b  in  terms  of  a  in  general 
valid  for  the  tenth-degree  equation  for  a. 

In  this  case  k  =  l  if  we  solve  F{c,  m  :  k,l)  =  o  for  c,  we  have  a  cubic  for  c  if  a=t=6. 

The  rational  expression  (80)  for  a  becomes 

[4ck-mic- 1)  {c^-2c-i)]^^{2c=i=i)k+m(c+i)c'^i2c-iy 


m(c-i)(3c+i)  /■-,  1  -M  1  ~^,-L,^,2      

=  5,  a  is  indeterminate,  the  limiting  valu« 
(§5)- 


m(c-i)(3c+i)  (2c+i)*+ot(c+i)c»     (2c-i)» 

for  c  =  5,  a  is  indeterminate,  the  limiting  value  leading  to  the  infinite  triangle  previously  discussed 


For  the  isosceles  case 


whence  a  =  b  = . 

2 

For  the  case  a^b  however 


k^     (c-i)(c+i)' 
m  8 


m        (2C-I-1) 
and  the  expression  for  a  becomes  indeterminate.    The  limiting  value  gives 

(2C3-C»+l) 


a=  — 


(2C+I)(C-I)' 


V 


DO- 


c 


>o 


C" 


I 
II 


SURFACE  r  (o-,  a,  0)=o  ■  59 

whence 

,  2C'  ,      ,  '  2C'i2C'  —  c'+l) 

(2C+l)(c-l)  (2C+l)'(c-l)' 

This  value  for  ab  is  however  inconsistent  with  Diia,  b)=o  which  gives 

ab  = 


(2C+I)- 

Hence  the  rational  expression  fails  as  was  to  be  foreseen  from  the  group  theory. 

19.       THE    SURFACE   Fi<T,a,  /3)  =  o 

Since  a  is  single  valued  we  take  the  «  axis  vertical.  /8  and  o-  to  the  right  and  up  respectively 
in  the  plane  of  the  diagram.     (Fig.  14.) 

This  shows  the  ridge  lines  T  =  o,  Z>3  =  o  and  the  lines  Z),=o  and  <^— t— 1=0  marked  <f> 
where  the  discriminantal  cylinder  has  an  ordinary  intersection  with  the  surface. 

The  projection  of  the  asymptotic  cylinder  a  =  00  on  the  /3,  o-  plane  is  marked.  The  cross- 
sections  give  a  descriptive  idea  of  the  surface.     (Fig.  15.) 

Drawing  to  scale  is  unfortunately  impossible  as  the  small  loop  of  the  asymptotic  cylinder, 
only  extends  to  ^=  .016  .  .  and  the  real  region  commences  at  18=27  where  the  ridge  lines 
have  a  triple  tangent. 


II 

I.      THE   EXTERNAL   PROBLEM 

The  formulas  for  the  external  bisectors  being 

^^^(^b+c)ia+b-c)bc 


{b-cY 

{c—a 
{—a-\-b-\-c){a—b-^c)ab 


{c—a) 

(  —  nJ-h 


{a-by 

as  in  the  internal  case  we  use  ratios  and  write 

K':L':M'::l:^:- 
k    I     m 

a+b+c=i 


}il-\-lm-\-mk  klm 

Expressing  a,  ft  in  terms  of  x,  y,  z  elementary  symmetric  functions  of  the  sides  and  writ- 
ing it;=i  we  have 

Uy'-y-3zy 

o.  ^ 

4y*—y^—6y'z+gz'—syz+z 

Q^ -(4/-y-32> '^^^ 

z{^y—%z—i){—^y^+y-\-iS,yz—2']z^—^), 

The  cubic  expression  in  the  denominator  of  /3  is 

P*=[{a-b){b-c){c-a)]' 

the  discriminant  of  the  cubic  whose  roots  are  the  sides. 
We  notice  also  that 

Da 

J^  /    \ 

4y* —y3—  6y'z  +  92' — 3)'2 + z 

Hence  all  isosceles  triangles  have  a =4,  /3=  00  . 

Points  on  0=4  for  which  /34=oo  are  reached  only  in  the  y  2  plane  along  43)'— 32  =  0  as  limits 
for  y=  00  ,  which  gives  infinite  sides  with  complex  approach. 

As  in  the  case  of  the  internal  problem  it  is  convenient  to  eliminate  in  two  ways. 


Writing 


2.      THE   FIRST  ELIMINATION 


60 


THE  GROUP  OF  THE  EQUATION  6l 

we  obtain 

4-°^3(4P<^+3'^+/>+i)  \ 

y3(4-«) p(<T-p)(p-a+l)'  i^^' 

8io        tr(— 9p»+9po— 9p+8(T)(3or— 3p+i)  / 

From  the  first  of  these 

,        (3tr+l)(l-a) 
3"=         (I  +  -)         ' 

and  by  substitution  in  the  second 

^{a-4)^{a+iy(i+ao-)[~a'(a+iy+a{ga'+io<T+s)-4]  .      » 

This  equation  of  the  first  order  in  /8  with  no  common  factor  of  the  coefiicients  of  /S"  and  /S" 
is  necessarily  irreducible  in  R{a,  /8). 

Since  all  subsequent  operations  must  depend  either  for  their  necessity  or  their  effectiveness 
on  the  nature  of  the  group  of  the  equation  it  is  proper  to  determine  this  in  advance  if  possible. 

3.      THE   GROUP  OF   THE   EQUATION 

For  a=o  the  equation  (6)  reduces  to  /8o-((r -f-i)'=i  which  has  a  binomial  approximation 
5*=j8  for  0-=-  at  5  =  0,  /3=o,  and  hence  a  cyclic  substitution  of  order  three  among  the  roots. 

27  ■ 

At  y3= ,  (T=  —\  3.  double  root  occurs  giving  a  two-cycle  and  establishing  the  symmetric 

4 
group  on  these  three  roots  as  a  subgroup  of  the  monodromie  group. 

For  /8=o  the  equation  reduces  having  a  pair  of  squared  factors.  Further  equalities  occur 
at  a =0,  I,  3,  4.  These  facts  may  be  represented  in  a  graph  where  the  upper  curve  is  double 
(Fig.  16). 

For  a4=o,  1,3,4  a  proper  approximation  is  of  the  form  «/?=  (o-— a,)"  where  <r,  is  one  of  the 
doubled  roots.  The  two  values  of  o-,  lead  to  a  substitution  of  the  form  (12)  (45).  At  a=i, 
o-=  —  I  and  at  0=3  0-=  — |,  the  crossing  points  of  the  curves,  no  three-cycles  occur,  but  at  0=4 
the  approximate  forms  are  3(125'+ a)'=— 4  Pas'  forcT=o+5and"  =  4+fl,  and  also  485'+  a  =  o  at 
0-=  —^+5.  These  give  a  substitution  of  the  form  (14)  (25)  (36).  The  two  substitutions  may 
be  denoted  by  U  and  V  respectively. 

For  a=3,  /3=27,  the  equal  bisector  point,  the  equation  reduces  to  a  perfect  sixth  power. 
The  approximation  is 

29165'— loa  where  <r=—\-\-s,  a  =  3+a,  P=2j. 

At  this  point  we  have  a  six-cycle. 

Without  further  specifying  the  identity  of  the  roots  we  may  now  prove  that  the  group  is 
the  symmetric  group  on  six  letters. 

Assuming  that  the  six-cycle  is  (123456)  the  three-cycle  falls  in  this  either  with  an  adjacent 
pair  or  in  alternate  positions.    That  is  either  i,  2,occurin  (a,6,  c)  or  (a,  6,c)  is  (135). 

In  the  first  case  transforming  (12)  by  the  six-cycle  we  get  (23),  (34),  (45),  (56),  (61)  and 


62 


THE   PROBLEM   OF   THE   ANGLE-BISECTORS 


compounding  with  (12)  successively  (13),  (14),  (15)  and  having  now  all  single  transpositions,  all 
the  substitutions  of  the  symmetric  group  follow. 
In  the  second  case  we  have 

5=  (i 23456)  :r=  (13s)  and  also  (13),  (35),  (15). 

By  using  5  as  a  transformer  we  add  (246) :  (24) :  (46) :  (62)  and  the  roots  fall  so  far  into  two 
disconnected  sets.  The  cube  of  S  is  (14)  (25)  (36)  and  V  is  of  the  same  form.  If  the  two  are  not 
identical  V  either  keeps  the  same  division  or  connects  the  odd  and  even.  In  the  latter  case  the 
transforms  of  the  transpositions  already  at  hand  give  all  transpositions,  in  the  former  cases  U 
can  be  used  as  a  transformer  on  V  and  the  same  result  is  reached,  namely  the  symmetric  group. 
As  in  the  previous  problem  the  algebraic  group  is  also  the  symmetric  group. 


The  equation  in  <t,  although  convenient  for  determining  the  irreducibility  and  the  group,  is 
under  disadvantages  in  other  respects.  The  expressions  for  y,  z  are  rather  complicated  and  fail 
to  give  determinate  values  in  rather  a  large  number  of  special  cases.  These  occur  for  the 
following  values: 

a=o  /34=o  when  three  roots  are  infinite  but  p  has  the  factor  i-|-a<^  in  its  denominator 
and  is  indeterminate. 

a=  I  /34:o  when  o-=  —  i  is  a  fourfold  root  and  p  is  indeterminate. 

a=3  when  o-=  —J  is  a  threefold  root,  p  indeterminate. 

It  is  not  necessary  to  conclude  that  a(a— i)(a— 3)  is  a  factor  of  the  discriminant,  for  the 
definite  values  of  o-  which  arise  point  out  a  discriminantal  point  rather  than  a  discriminantal 
locus.  ' 

To  avoid  these  things  which  cannot  all  be  successfully  dealt  with  by  limits  a  second  method 
of  elimination  is  convenient. 


THE   SECOND   ELIMINATION  63 

4.     THE    SECOND   ELIMINATION 

Writing 

P±Z4)^B:^-^=A  •  (7) 

a  a  ' 

we  have 

y-4y'+3z  _R  xjjx 

zUy-8z-T)-^  ^^> 

(3>'-i)[z(6y-i)-/(43'-i)]_  .  /oM 

These  are  quadratics  in  z.  The  elimination  is  simplified  by  writing  z=(4y'—y)t  and 
eliminating  t,  after  division  by  (43^ — y) .  The  values  y  =  o,y  =  \do  not  in  general  give  solutions, 
with  2  =  0. 

Arranged  in  powers  and  products  oi  A,  B  the  result  is 

F{y,A,B)=     A'B'Uy-inSy-sVy 

+AB'   [4y-i]^[3y-i][i28/-96y+i7]3' 

-gAB  Uy-iVlsy-'^] 

+8B'   [4y-iY[3y-^]'Uy-i]y 

-2yA    Isy-i]'' 

—SB      [33;-i]3[6y-i]=o 


(9) 


For  «  =  3,  18=27  •  ^  =  ~l>  B=—g,  this  reduces  to 

36(2^—1)^=0 

giving  a  unique  complex  triangle  with  equal  external  bisectors. 
Here  y  =  \,z='h)  ^^^ 

a  \  b  :  c=2-\-^\/2 — ^1/4  :  2+w  3y  2  — «)^  •S]   4  :  2-\-oi'  ^y.  2  — <o3^/  4 

where  <o=(-^+'^'^). 
2 

That  this  equation  is  irreducible  and  has  the  general  group  follows  from  the  fact  that 

y  is  a,  rational  function  of  <t,  a,  j8,  while  A ,  B  are  rational  functions  of  a,  fi. 

Expressed  as  a  rational  function  of  {y,  A,  B),  z  is  given  by 

(^y-y)[SABUy-i)y+&Bi3y-i)y+gA]  ,     . 

3AB[64f-28y+s\+SB[i8y^-gy+i]+2'jA  ^     ' 

This  is  the  form  obtained  at  the  penultimate  step  of  the  elimination  and  has  the  disadvan- 
tage of  being  indeterminate  for  27^  =B,  y  =  \  in  which  case  the  form 

o-.. .,    ,  I    3^(4y-i)(8y-3)+8(3y-i)'  ,. 

S2-4y     1+ 8^5(4y^-y)+85(3y-y)+9^  ^"^ 

or  2= J  for  the  special  values,  may  be  used.     This  is  identical  in  virtue  of  F{y,  A,  B)=o. 

The  rational  integral  expression  for  z  in  y,  ^4 ,  B  appears  to  be  too  complicated  for  use. 


64  THE  PROBLEM  OF   THE  ANGLE-BISECTORS 

5.      MULTIPLE    ROOTS 

The  method  is  that  of  the  internal  case  (§  10).     If  a-{-b-\-c=i  and  z  =  abc,  we  have, 

z  taking  the  place  of  <  as  a  parameter  in  each  equation  of  the  set  of  three  but  not  being  a  propor- 
tionality factor  as  in  the  internal  case,  the  algebra  becomes  a  little  more  complicated. 
For  variations  near  a  solution  we  have 

ia\—\a?-\-iia}—2a{x  —  2kz)-\-z{^\  —  li)\ 
-\-dk\2a^  —  a-\-\'^z  \    (13) 

-\-dz\2a'k-\-a{s,—K)-\-'&]i'^  —  Q 

At  ordinary  points  there  are  three  equations  of  the  form 

da=Adz+Bdk, 

db  =  Cdz+Ddl,  ]    (14) 

dc=Edz-\-Fdm 

As  in  the  internal  case  we  conclude  that 

A+C+E=T 

is  a  factor  of  the  discriminant  and  also  that  k  =  l,  0^  =  0  gives  rise  to  double  roots.    Whether 
or  not  D2  recurs  as  in  case  of  the  internal  problem  is  more  troublesome  to  determine  on  account  of 
the  complexity  of  the  expressions,  and  the  question  is  postponed. 
As  before  we  investigate  these  discriminantal  factors 

~     -4a5-l-7a''-|-i6a32-4a3-|-(i6z-|-i)a"-8az+i62»  ^^' 

Expressing  the  summand  as  far  as  possible  in  terms  of  y,  z  by  means  of 

a^—a^-\-ay—z  =  o 
^^  2  [(2/-8yz-z)a'-f(43'z-y'-8z''+2)a+(y2-6z'')]  ,^. 

~        i(y-4z)a'-f-(i6y2— 4/+y-S2)a-f43'2-z)] 


(17) 


and  the  explicit  evaluation  of  the  symmetric  functions  leads  to 

z''[— 6i44y^-|-28i6y— 32o]-(-z3[4og6y3— 24ooy^-|-444y— 25] 
-|-z^[  — 896y''-f  6o8y3— i36y^-f  ioy]4-2[64y^— 48y''-|-i2y3— y^]  =  o 

which  reduces  to 

r=z(4y-i)[4>'(>'-62)-y-Sz)][4(y-4z)='-(y-52)]  =  o  (18) 

For  convenience  we  call 

4y(y-6z)-(y-sz)  :  T,, 
4(y-42)^-(y-52)  :  T,. 

As  in  the  internal  case  the  vanishing  of  the  denominators  must  also  be  discussed. 
The  condition  that  the  numerator  and  denominator  for  a  should  both  vanish  is 

rj[z»(32y-z)+z(i28y3-64/-y+z)-f(-32'i+2oy3_3y^)]  =  o  (19) 


MULTIPLE   ROOTS  65 

The  second  factor  is  however  an  extraneity  as  it  does  not  also  consist  with  o^— a^+flj'— 2  =  0. 

Tj  may  simply  be  expressed  by  ^=  — 4  or  as  a(a— i)+4z  =  o. 

The  latter  relation  multiplied  by  the  corresponding  expressions  in  b,  c  and  expressed  in 
y,  z  is  z^Ti-o,  while  if  k  be  replaced  by  —4  in  the  equation  for  a  (12)  this  becomes  the  square 
of  a(a— i)+4z=o. 

T2  =  ois  not  however  a  proper  discriminantal  factor,  for  the  value  ^=  —  4  has  no  relation  to 
the  homogeneous  problem,  and  if  a+b+c^  i  the  a  equation  does  not  become  a  square.  More- 
over the  expression  for  8k  becomes  a  perfect  square  for  ^=  —  4  and  we  do  not  obtain  two  dis- 
tinct solutions  in  the  neighborhood. 

Ti  in  the  a,  j8  plane  is  represented  by  /8(4— a)  — 8a  =  o,  in  which  the  factor  F'  enters  so  that 

the  triangles  given  along  Ti  =  o  are  in  a  sense  associated  with  isosceles  triangles.    None  are 

real  or  possible. 

(±y^ — y) 
The  factor  T,  is  purely  extraneous.     If  r,=o,  ^=(—zf~\  >  which  is  satisfied  by  a  =  §, 


■■\,c=\  :  y=i'6,2  =  BV 
For  these  values 


Namely  for  T,  =  o 


/3=co  ,  a  =  4  but  |8(a  — 4)  =  oo. 

4(i6y-s) 


a-4=- 

T  ' 


64/ -5 
128(3^-0(24^-5)3; 


(i6y-s)^ 

For  this  set  of  values  the  equation  for  o-  has  roots  o,  o,  —  i,  —  i,  —  j,  and  —  |  and  the 
last  named,  a  single  root,  is  the  value  giving  the  triangle. 

Taking  up  the  factor  z  we  have  for  2  =  0, 5=  =»  ,  A  =  —  .     _   .  if  y+o,  y^j.     For  B=  00 

the  equation  Fiy,A,B)  =  o  becomes  (4^- i)^3;[i63'3(4.4 +3)^- 8^^(32^^4-5271  + 2i)-f-y(84^"-|- 

147^-1-64)— (9/l-t-8)(yl-|-i)]  =  oandy=  ,  ,  is  a  single  root.     Hence  z  =  o  is  not  discrimi- 

(4^+3J 
nantal  unless  ^'=5  or  y  =  o  and  these  points  occur  on  the  other  factors.    The  factor  2  is  then 
extraneous. 

We  are  left  with  (4^—  i)  which  is  in  fact  a  discriminantal  factor  and  will  be  referred 
to  as  T.    In  the  A,  B  plane  it  is  represented  by  27^1—5  =  0  and  in  the  a,  ji  plane  by 

(a-4)(/?+27)  +  8l  =  0. 

Using  a  similar  notation  to  the  internal  case  we  write  D2  =  o  as  the  representative  of  the 
equal-bisector  non-isosceles  locus. 

11  k  =  l,  a^b,  there  are  in  the  {a,  b,  c)  plane  three  factors  of  the  form 

D2{a,b)  =  2{a-\-by-\-i^ab{a-{-b)-s{a-\-by-2,ab-\-4r(fl+b)-i  =  o. 

Expressing  the  (a,  b)  form  in  c  and  z  =  abc  if  a-\-b-\-c=  i  we  have 

2C''— c3-(-4C2— 2  =  0. 

Reducing  by  c'  — c'-f-yc— 2  =  0,  we  have  c  =  o  or 

31  —  62 

c——- . 

231—1 


66  THE  PROBLEM  OF   THE  ANGLE-BISECTORS 

Eliminating  c 

Z?2(3',z)  =  i6)'32— 4}'^+2i62'— i8o3's'+3oy'2+y'+362^— i23'z+2  =  o  (20) 

This  factor  of  the  discriminant  has  of  course  the  same  representation  in  the  a,  jS  plane  as  in 
the  internal  case. 

For  D2  =  o  the  sextic  becomes  a  perfect  square.     If  ^  =  /=  i, 

£  =  w»-2W-8,  A  =  -^    ,     ,     ',      . 

(w+2)3 

With  these  values  F{y,A,B)  becomes 
[i6>^(w— 4)^w— 8/(m— 4)(w^— 8w— 2)-)-j'(w3— 2iw'+75W+S3)+(w'— 8w— ii)p      (21) 

In  the  reduction  the  factor  (w+2)''  is  removed  from  both  numerator  and  denominator. 
For  m=  —  2,  0  =  0,  /8=o  and  the  affair  is  indeterminate.  For  this  case  however  A  =  ^  ,B  =  o 
with  AB=  00  ,  AB^  finite,  and  the  limiting  values  serve. 

We  conclude  from  the  set  of  three  pairs  of  equal  roots  that  as  in  the  internal  case  D\  is 
a  probable  factor  of  the  discriminant. 

An  expression  for  the  discriminant  in  the  (^,2)  plane  may  be  obtained  by  equating  the 

values  of  the  derivatives  —  as  given  by  the  two  equations  (8)  (8')  and  eliminating  A ,  B.    This 

dz 

process  gives 

D{y,z)  =  {Ay-T)Uy^-y-2>zy{D2{y,z)]  =  o. 

Hence  as  ^■f—y—2,z  vanishes  only  for  a  =  o,  ;8=o  no  new  factors  are  obtained  by  this 
method. 

6.      THE  NODAL  CURVE 

Among  the  factors  of  the  discriminant  of  F{y,A,B)  is  one  which  relates  to  equaUty  of  the 
y's  only  without  implying  equality  of  the  2's  and  hence  not  discriminantal  for  the  problem. 
This  arises  from  the  nature  of  the  elimination  process:  the  two  quadratics  from  which  z  was 
eliminated  may  become  identical.  There  are  two  conditions  in  y,A,B  from  which  y  can  be 
eliminated.    The  result  is 

AB{a-A)P=o 

where  P  expressed  in  a,  ^  is 

P  =  ^'(a-4)(a-9)  +  54^(a-l)(2a-9)-|-729a(a-i)  (22) 

When  P  =  o,  y  is  given  by 

y=[|8(a-4)(a-9)-27a(a-i)]+[l2/3(a-4)(a-3)]  (23) 

Since  a=  oo  does  not  cause  ^  or  5  to  vanish  the  effective  factors  are  (i  — a)(a— 4)»P. 

The  line  a=  i  is  in  fact  a  nodal  line  on  the  surface  F{y,A,B)  =  o.  P=  o  is  also  a  nodal  line 
while  a— 4=0  gives  an  infinite  cuspidal  edge. 

The  last  locus  is  discriminantal  for  the  problem  while  the  others  are  merely  so  in  relation 
to  the  choice  of  y  and  the  elimination  method. 


THE  DISCRIMINANT  67 

7.      FINITE  MULTIPLE   POINTS   OF   ORDER  HIGHER   THAN   THE   SECOND 

If  A  =  o,  r=t=o  and  if  no  other  possible  factor  of  the  discriminant  vanishes  the  only  possi- 
bilities are  4-points  and  6-points. 

The  discriminant  of  the  cubic  factor  to  the  square  of  which  F(y,A,B)  reduces  for  D2  =  o 
is  to  a  numerical  factor 

(w— 4V(ot—  i)*(w'+wj+7). 

There  should  also  be  counted  m=<x  (tt=oo,|8=oo). 
For  m=  I  :  1=3,  ^=  27  occurs  the  6-point  y=^  already  discussed. 
For  »»=4  :  1=4,  )8=any  finite  value,  four  values  of  y  are  infinite. 
The  only  real  finite  y8  on  D2  =  o  and  on  a  =  4  is  54. 

The    complex    factors    are    intimately    connected    with    ot  =  4   in    that    if    w= , 

3 

(w— 4)(w'+w+7)  =  27(»3— i).    They  give 


a 


_i±VjZl 


y3=i\(-i4±i/-3) 


and  are  intersections  of  T  and  D. 

For  r'=o  we  have  y=\  which  is  a  double  root  for  all  ^'s  when  2'jA=B  and  a  fourfold 
root  when  34'+3^  +  i  =  o.  These  values  give  the  complex  4-points  just  mentioned  and  there 
is  never  a  6-point. 


The  list  of  multiple  points  is 

then 

''=3, /3=27 

a  6-point 

on  D2,  P,  not  on  T 

„_(3*l/-3) 
2 

- 

two  4-points 

on  Da  and  T 

a=4,  /3any 

a  4-point  line 

a=o,  /8=o 

is  indeterminate 

a 

a  3-point 

y=i  on  T 

a' 

a  4-point 

y=  00 

a=oo,  j8=  00 

a  double  3-point 

y  =  o,y=\ 

a=oo,  j8=o 

a  4-point 

y=  00 

a=0,  /3=oo 

is  indeterminate 

(a  3-point  if  JB  is  finite) 

8.      THE   DISCRIMINANT 

The  following  factors  have  been  found: 

AU,fi)  =  i5'(A  +  i)»-|-25(9^+7)+27(4^-f3) 
'      ■  i'U,£)  =  B^(94+8)-S4^B(9^  +  7)-729^(4^-f3) 

T{A,B)  =  2^A-B 

A;  B;  4^+3;  and  J?  =  00  is  also  discriminantal. 


68  THE  PROBLEM   OF   THE   ANGLE-BISECTORS 

If  any  other  discriminantal  loci  exist  they  must  have  common  points  with  either  ^  =o  or 

For  A=o  54=0  the  equation  becomes 

-B(43'-i)'(2>'-i)3'-(3>'-i)(6>'-i)=o 

after  dividing  out  the  pair  of  factors  (sy—iY  which  belong  toA=o.  Further  equalities  occur 
ioT  B  =  o,  B  =  co  and  for  values  of  y  which  are  roots  of 

288y  —  288y'+ 104^— 163'+ 1  =  o, 

the  corresponding  value  of  B  being  given  by 

5(32/  — i6y+i)  =  9. 

The  intersections  of  ^4  =0  and  the  locus  D2  give  for  B  the  quadratic 

5^+145+81=0. 

In  the  field  of  the  complex  roots  of  this  equation  the  quartic  for  y  reduces  and  gives  the 
set  of  four  y's,  two  for  each  value  of  B,  and  as  D  is  the  locus  of  three  pairs  of  equal  roots  we 
have  no  outstanding  discriminantal  points  on  ^  =  o. 

For  B  =  cc  the  equation  becomes 

^»(4y-i)(8y-3)^+yl(3>'-i)(i28/-96>'+i7)+8(33)-i)='(2>'-i)=o, 

after  the  pair  of  factors  belonging  to  .6  =  00  [(4^—1)^]  and  the  factor  y  have  been  set  aside. 
The  values  of  A  for  which  y  is  a.  double  factor  are  —  i,  and  —  §.  The  former  belongs  to  D2 
and  the  latter  to  P,  and  the  system  of  equalities  is  in  each  case  what  is  required  of  such  inter- 
sections.   The  discriminant  of  the  cubic  in  A  is 

64^^(4^+3)4U  +  i), 

and  since  the  term  A^  is  absent  A  =  <x>  must  be  discussed. 

For  A=o  one  pair  of  equal  roots  occurs  in  keeping  with  the  discriminantal  character  of  A . 
A=  —I  belongs  to  D,  and  has  in  all  four  roots  —\  and  two  y  =  o.  This  is  in  accord  with  the 
fact  that  D2  has  normally  3  pairs  and  B=oo  is  simply  discriminantal.  A=  —J  is  itself  a 
factor  and  the  two  pairs  occurring  are  expected.    A  =  x  has  no  extra  equalities  for  B  =  <x  . 

We  conclude  that  the  factors  occurring  in  the  discriminant  are  A,  4^+3,  B,  T,  P,  D,; 
that  ^  =  oQ  and  B  =  oo  are  discriminantal  and  that  there  are  no  others. 

The  complete  discriminant  is  then  of  the  form 

N  •  A"  •  B''  ■  (4^1+3)"*  •f'-Pi'-  D-',. 

To  determine  the  exponents  and  the  numerical  factor  N  special  values  of  Bezout's  determinant 
are  calculated. 

First  put  A.=  —I,  a  non-discriminantal  value,  and  5  =  27.    The  computation  is  rendered 

comparatively  light  by  transforming  y  to and  multiplying  up  by  16  to  obtain  an  integral 

4 
form.     This  divides  the  discriminant  by  2'°.    Writing  B  =  2']x  and  dividing  by  27  we  have  the 
form 


INTERRELATIONS  OF   THE   TWO   EQUATIONS  69 

The  value  of  the  Bezout  determinant  for  this  form  is  if  a:=  i 

2'5  .  3^4  .  53  and  if  x  =  2,  2^"  •  3^'  •  13^ 

The  algebraic  factor  of  T  is  x+i,  of  P,  (r'+4.r+i),and  of  D„  {^x+i).  From  these  values  we 
conclude  that  d  =  s,  />  =  2,  /=  i,  i  =  8.  The  values  of  a  and  m  cannot  be  determined  since  both 
A  and  (4^4+3)  are  (—1). 

Replacing  the  factors  divided  out  in  the  transformation  we  have 

A=  ± 23^  •  35"  .  yl <■•£»•  r  •  P"  •£>» (4^+3)". 

To  determine  a  and  m  we  give  A  and  B  simple  non-discriminantal  values  and  calculate 
the  residue  of  the  Bezout  determinant  modulo  a  suitable  number  prime  to  all  the  factors  of  A. 

A^i,  B  =  i  gives  4  •  7"=  — 2  (mod  11) 
i4  =  2,  5=1  gives  2"+'    =  —  2  (mod  5) 
^  =  2,  B  =  i  gives  2"+'"  =     1  (mod  7) 

That  is  provided  the  positive  sign  is  taken  with  A.  The  only  solutions  permissible  on  account 
of  the  limitations  of  the  order  are  a  =  2,  m=2.  There  is  no  permissible  solution  with  the 
negative  sign  for  A. 

On  account  of  the  connection  in  general  between  the  Bezout  form  and  the  standard  form 
of  the  discriminant  the  result  is  to  be  divided  by  —6"  and  we  have  finally 

-2^8  .  2^»  .  A'  ■  B^  -f  •  P'  ■  DlUA+sY 
as  the  value  of  the  discriminant. 

Q.      THE   INTERRELATIONS    OF   THE    TWO   EQUATIONS 

The  variables  y  and  o-  are  connected  by  a  birational  transformation,  namely 
A  +  i+<T         _  8ABUy'-y)+SBisy-i)+gA       i-jy 


'4A+3-3T'  B[3AUy-i)iSy-3)+i3y'—iy]        y 


(24) 


In  general  the  discriminantal  factors  will  be  identical  but  at  certain  critical  points  and  lines 
irregularities  may  occur. 

For  A=  —  f  (a  =  3)  the  equation  in  a-  has  three  roots  equal  to  —  ^  and  the  expressions  for  y 
become  indeterminate.  The  proper  values  for  y  are  in  this  case  the  three  roots  of  the  expression 
for  o-  considered  as  a  cubic  in  y.  In  fact  the  locus  A=  —  f  is  not  discriminantal  for  the  y  equa- 
tion but  a  locus  of  reducibility,  the  reduced  factors  being 

5(4>'-i)'(y-i)-9  (3y-i)  and  J5(8)'3-i2>^-t-3y)-9(3y-i)  (25) 

In  a  similar  way  for  ^  =0  the  o-  equation  has  four  roots  equal  to  —  i  and  the  expression 
for  y  is  indeterminate,  and  that  for  o-  fails  to  give  the  y  values  which  must  be  sought  from  the 
y  equation  which  has  distinct  roots  for  these  four  places. 

Similar  irregularities  occur  for  B  =  o,  jB  =  00  ,  and  A=<x> .  They  are  complicated  by  the 
fact  that  the  connection  between  {A,  B)  and  (a,  fi)  is  also  birational.  These  do  not  call,  how- 
ever, for  any  special  computations. 


7° 


THE   PROBLEM   OF   THE   ANGLE-BISECTORS 


lO.      THE   TRANSFORMATIONS  . 

To  avoid  the  birational  transformation  from  a,^  to  A,B  and  to  keep  as  close  to  the  tri- 
angles as  possible  we  consider  the  transformations  leading  in  a  chain  from  the  sides  {a,b,c)  to 
the  symmetric  functions  of  the  sides  {x=i,y,z)  and  to  the  symmetric  functions  (a,/3)  of  the 
angle-bisectors  which  may  be  taken  as  the  data  of  the  problem.  The  equation  F{y,A,B)  =  o 
will  then  be  considered  as  if  its  coefficients  were  explicitly  written  in  (a,  /i). 

We  trace  the  discriminantal  loci  in  the  {a,b,c)  plane.  Beginning  with  tt  =  ao ,  the  repre- 
sentative is 

^ab{b-cy{c-a)'ia-b+c)  {-a+b+c)  =  o. 


This  is  a  curve  of  the  eighth  order  with  sixfold  symmetry.  It  is  not  difficult  to  establish  the 
following  features.  The  curve  has  no  real  infinite  branches.  The  center  of  the  triangle  of 
reference  is  a  conjugate  point.  The  curve  has  two  branches  at  each  vertex  touching  the  sides. 
It  has  two  branches  at  the  mid-points  of  the  sides  touching  with  inflexion  the  lines  a+b—c  =  o, 
etc.     The  extent  to  which  the  branches  leave  the  sides  is  determined  sufficiently  by  the  points, 


a=.205 

b=    .045 

c=   -75 

a=  .46s 

b=-.2is 

c=    -75 

a=  .06 

b=-.26 

C=1.2 

a=  .21 

6=  —  .41 

C  =  1.2 

and  the  symmetric  correspondents  (Fig.  17). 

The  locus  i8=oo  has  three  representatives:  (a  — 6)(J— c)(c— a);  abc,  a.nd  (a+b—c){a  —  b+c) 
i-a+b+c). 

The  loci  a  =  o  and  /8=o  are  jointly  represented  by 

ta{b-cy{b+c-a)  =  o. 


THE   TRANSFORMATIONS  71 

This  curve  has  sixfold  symmetry  and  is  closed.     It  may  be  traced  by  writing  a  =  x+y,  b  =  x—y, 
c  =  i  —  2x  when  it  takes  the  form 

2y*+-fii2x'—iix+2)  +  {i8x*-2ix'+&x'-x)=o. 

As  a  quadratic  in  y"  the  discriminant  is 

—  96a;3+iosx"  — 36X+4, 

whose  one  real  zero  x  =  .  549  .    .    .  limits  the  curve  to  a  triangle  slightly  larger  than  the  refer- 
ence triangle. 

The  curve  passes  through  the  vertices  parallel  to  the  opposite  sides  and  meets  the  sides 
also  at  the  midpoints  touching  them  there.  It  has  no  singularities  except  a  conjugate  point 
at  the  center  of  the  reference  triangle. 

The  discriminantal  factor  T  is  represented  by  4{ab-\-bc-i-ca)  —  1=0.  This  is  the  inscribed 
circle  of  the  reference  triangle.  The  factor  D,  is  represented  by  three  symmetrical  curves  of 
which  Diia,  b)  is 

2ia+by+4ab{a+b)-5ia+by-3ab+4{a+b)- 1=0  (26) 

Writing  a-\-b  =  2x,  a  —  b  =  2y  we  have 

(8a;'-s^+i)(3a;-i)  . 
^  8x-3 

y  is  real  except  for  ^\<a;<^\. 
The  asymptotes  are 

y=|/3a;— Tj\i/3  and  the  conjugate. 

The  intersections  with  the  asymptotes  are  at  a;  =  iVi!. 

The  curve  passes  through  the  center  of  the  triangle,  the  midpoints  and  the  vertices  where 

it  touches  the  sides.    At  the  midpoints  the  direction  is  »t  =  —  f  (Fig.  18). 

By  determining  these  curves  and  their  intersections  the  diagram  of  a  one-sixth  part  of 
the  (a,  b,  c)  plane  symmetrical  and  serving  as  a  fundamental  region  is  obtained.  It  has  sixteen 
compartments  (Fig.  19): 

Regions  (i),  (2)  corresponding  to  real  possible  triangles  with  three  external  bisectors, 
regions  (3),  (4),  (5),  (6),  (7),  (8),  (11),  (12),  (13),  (14),  corresponding  to  real  impossible  triangles 
with  pure  imaginary  bisectors,  regions  (9),  (10)  with  real  possible  triangles,  one  bisector  being 
external  and  two  internal,  and  regions  (15),  (16)  with  impossible  triangles  and  real  bisectors. 

After  the  symmetric  function  transformation  these  compartments  are  to  be  followed  to  the 
(y,  z)  plane. 

The  y,  z  Plane 

The  locus  a=co  is  represented  by 

4y*—y'-\-gz'  —  6y'z  —  2yz-^z  =  o. 


Real  points  only  occur  in  general  for  |  y  |  <  i|/  3.    There  is  however  a  conjugate  point  at 

y  =  ^,  2  =  ^;.    z  has  a  maximum  for  y  = K^        ,  z- 

conjugate.    At  the  origin  there  is  an  inflexion  y'  =  z. 


y  =  i,  2  =  V--    2  has  a  maximum  for  y  = ~ — ^,  z  =  — '^ ,  and  a  minimum  at  the 

J'     3,        2.  -^  16        '  576 


72 


THE   PROBLEM   OF   THE  ANGI-E-BISECTORS 


The  sides  and  bisectors  of  the  reference  triangle  transform  as  in  case  of  the  internal  prob- 
lem, the  real  region  being  inside  the  curve  A  (Fig.  20).  The  closed  curve  a  =00  lies  entirely 
inside  the  curve  Z),. 

a  =  o,  /8  =  o  is  represented  by  the  parabola  32— 4y-|->'  =  o.  This  passes  through  theorigino, 
the  point  F(j,  o),  the  point  A (^,  ^jV)  and  cuts  the  line  4y  —  8z  —  i  =o[/3=  00  ]  at  F  and  Q(|,  i\) 
and  has  no  contacts  with  the  other  curves  in  the  real  region  {y,  z). 


Along  Z?,  ()8  =  00  ,  a  =  4)  is  represented.    3  =  0  represents  /3  =  00  and  so  does  /^y  —  8z  —  i=o. 
The  curve  D2  is 

i6yH  —  ^y*+2i6z^-iSoyz^+Soy^z+y^+2,(>z'—i2yz+z  =  o  (27) 

This  approximates  the  semicubical  parabola  2 72^+ 2^3  =  0  in  the  infinite  regions,  and  has 
an  asymptote  323)— 1282  — 5  =  0,  the  further  intersections  with  which  are  complex.  There  is 
a  conjugate  point  at  (^,  iV)  on  the  ordinary  branch,  an  inflexion  at  the  origin  yi+z=^o,  and  no 


THE   TRANSFORMATIONS 


73 


other  singularity.    It  touches  Z?,  at  (J, o)  having  43/— 8z—  i  =0  as  the  tangent  and  also  touches 
Z),  at  the  cusp  A .     From  the  cusp  out  to  y=cc  D^is  outside  Dj,  otherwise  inside. 

T  is  represented  by  the  line  y  =  4 . 

There  is  no  difficulty  in  identifying  the  regions  (i)  to  (16)  in  the  {y,z)  diagram.    They 


Case     n.U.Vcy 


cover  the  interior  of  D,.    The  exterior  is  divided  by  the  curves  into  twelve  regions  which  are 
marked  I,  II,   .    .  XII  as  a  basis  for  discussing  the  transformation  to  the  (a,  j3)  plane. 

Tke  a,  /3  Plane,  Limits 

Since  the  transformation  from  {y,  2)  to  (a,  y8)  is  not  everywhere  point  for  point  it  is  neces- 
sary to  investigate  certain  limits. 


74 


THE   PROBLEM   OF   THE   ANGLE-BISECTORS 


GC        „                             \ 

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THE   TRANSFORMATIONS  75 

Limits  for  infinite  values  of  y,z  are: 
Along  z  =  my' 

For  tn=(X} :  a=  i,  /8=o 

for  w  =  o :  a  =  4  but  j8  =  o  is  not  reached,  z  =  o  giving  /3  =  oo  .  The  point  a  =  4,  /3  =  o  is  attained 
as  nearly  as  we  please  in  VIII,  IX  and  also  in  (15),  (14).  Any  positive  o<4,  ^  =  0  is  reached 
in  IV  or  V  and  IX  and  also  in  I,  XI,  the  parabolas  being  eventually  outside  Z>,. 

Along  z'=my^ 

o  -8 

"~'^''^~w(27w  +  4)' 

0  =  4,  )8  any  negative  is  reached  in  I,  XI,  V,  IX.  At  w=  —^\  the  curve  eventually  falls  on  £>, 
but  /8=  —  00  so  that  no  limitation  is  set  on  /3  in  XI. 

For  the  limits  at  ^(§,  jV)  writing  y  =  ^+\  z  =  ^+p  we  have  . 

(5^-9P)'  27(5A-9p)^ 

"     i3X»-63Xp  +  8ip»        '^    3('^-3P)'+4'^'' 

Using  the  tangents  to  the  D^  curve  and  the  parabola  representing  a  =  o,  /?  =  o  as  axes 

5A.-9P  =  X 
X-3p=F 

and  approximating  along  semicubical  parabolas  Y'  =  mX^,  we  obtain 

For  positive  w's  o<  /3<  54.  For  o>w>— g/Sis  positive  and  the  curve  lies  in  regions  (i), 
(2).    For  »t< —i  the  curve  is  in  VII,  XII. 

Hence  VI,    VIII  cover  0=4  from  /8=o  to  54, 
VII,  XII    cover  a  =  4  from  /3=o  to  —  00  , 
(i),  (2)  cover  a  =  4  from  ^8=54  to  +00  . 

AtAii\=pp,a  =  — ^^~^^^l  ,,  and  j8=o. 
'^'^'        i3-63/>+8i/)' 

For  />  =  3,  a  =  4  and  for />  =  !,  a  =  o. 

Hence  values  of  a  between  o  and  4  are  found  paired  in  VII  and  VIII  and  also  in  XII  and  VI. 

Limits  at  F(i,  o)  are  investigated  by  writing  y  =  \-\-^,  z  =  p. 

We  have 

I6(3P-X)'                    (3P-^y    ' 
"■  =  — ; P  —  — 7a v;  • 

X— 2p  p(X— 2p)' 

Using  (X— 2p)  the  tangent  to  0=00  and  j8=oo  as  X  axis  and  X— 3p  the  tangent  to  a  =  o,  ^  =  0 

as  Y  axis. 

i6X»         „  X^ 


'Y'(Y-X)' 

For  Y  =  mX' :  a='^^  ,   /8=oo,  for  F=wX  :  a=o,  /3=--r-^ r  . 

m  m  \m  —  \) 


76  THE   PROBLEM   OF   THE  ANGXE-BISECTORS 

and  we  have 

»i>i  :  j8>o  the  region  VIII, ->3, 

P 

|<OT<  I  :  /8<o  the  region  IX,  -<o, 

P 

o<w<|  :  /8<o  the  region  X,  o<-<  2. 

P 

For  -  between  2  and  3,  m  is  negative  and  the  regions  are  the  real  sheets  (i),  (2),  (3),  (5), 

(6),  (7),  (12),  (11),  (10),  (9). 

Limits  at  g(f,i>e)- 

Here  4y  —  8z  — 1=0,  (/3  =  o)  and  32  —  4^^+^  =  0,  (n  =  o,  j8  =  o)  cut. 

If  y  =  j+K  z=A+p:'i  =  o,  ^=i6(2A-3p)3--2i7(X-2p), 

and  cubical  parabolas  give  a  =  o,  /3  any. 

/3  is  positive  in  V,  VII  and  negative  in  IV,  VI. 
Limits  at  0(o,  o). 

»-=(y+3zy-^{z-y'),  /3=(y+32)3H-(42-y")z 

are  proper  approximations. 

Along  z=my'  :  o=—  ,  /8=co  . 
pt 

These  parabolas  fall  in  III  for  w>j  :  /8>o,  a<4  and  do  not  fall  in  II  but  in  I,  which  is 
limited  by  a  =  4.    Hence  I  and  III  are  joined  at/8=oo,o<a<4. 

Along  y^  =  4l3z'  :  a  =  o.  These  curves  all  fall  outside  A,  and  join  I  to  (13)  for  a=oo  at 
O  is  y'-\-z  =  o  and  is  closer  to  the  axis  than  all  the  semicubical  parabolas. 

The  point  /(f,  i\)  is  a  conjugate  point  on  D.  and  ^  =  5  is  a  sixfold  root  for  a  =  3,  18=27. 
Two  sheets  from  the  real  y,  z  regions  are  VI  and  VIII :  the  other  four  sheets  are  complex  in  the 
neighborhood. 

In  transforming  to  the  (a,  /3)  plane  we  notice  first  that  for  the  whole  of  the  regions  outside 

A,  0<a<4. 

P' 
For  a— 4  =  -—  where  P'  is  the  discriminant  of  the  equation  whose  roots  are  the  sides  and 

is  negative,  while  Da  the  denominator  of  a  only  vanishes  on  a  closed  curve  inside  Di  and  is 
positive  outside:  while  a=  (32— 4/+;y)^-hZ}a  is  positive  (3)  (II,  i). 

The  a,  p  Plane 
In  this  plane  we  trace: 

D{a,  /3)  as  in  case  I. 

r(a,/3)  =  (a-4)(^+27)+8l=0  (28) 

P(a,^)  =  /3^(a-4)(a-9)  +  S4^(a-l)(2a-9)  +  729a(a-l)  =  o  (29) 

a  =  o,  a=i,  0  =  4,  /3  =  0. 

These  are  all  the  finite  discriminantal  lines. 

The  locus  P  =  o  has  no  representative  in  the  y,  z  plane  but  effects  there  a  pairing  of  points 
which  come  together  in  the  a,p  plane  forming  a  nodal  line  in  the  y,"-,^  surface.    The  locus 


V}—  ■^    S^fO^   XJJiJ 


78  THE  PROBLEM  OF   THE  ANGLE-BISECTORS 

a=i  is  of  the  same  character  while  0  =  4  and  y8  =  o,  which  are  infinite  cuspidal  edges  on  the 
surface,  partake  of  this  property. 

The  regions  (i),  (2)  being  within  D^  and  outside  a=oo  in  the  y,z  plane  have  a>4  /3?r54 
and  reach  0=4  at  A  for  any  /8^S4,  and  a=  00  ,  /3=  00  at  F.  Being  divided  by  A  they  fold 
on  D  and  on  a  =  4  and  cover  the  region  inside  D's  cusp  and  above  0  =  4. 

The  regions  (7),  (8)  are  within  Z),  and  have  a<o,  /8>o  and  join  on  D^. 

In  (a,  /3)  they  fold  on  D  in  the  fourth  quadrant  reaching  a  =  o,  /8  =  o  and  a=  —  00  ,  ^=  00  . 

The  regions  (3),  (4)  are  separated  by  T  in  (^,2;),  are  within  Dj  and  outside  a=  00  .  Hence 
a>4,  ;8<o.    Theyreacha=4,  ;8=oo  on  r  and  Z),  also  a=oo,  ;8=— 27  on  r  and  a=oo  {y  =  \, 

The  regions  (5),  (6)  also  double  on  T  but  here  a<o,  /8<o. 

The  regions  (15),  (16)  have  a>4,  /3>o  and  fold  on  D.  They  reach  a  =  4,  ^  =  54  along 
Di  at  infinity.  A  is  here  approximately  2']z'-\-2y^.  (15)  reaches  a  =  4,  P=  <»  along  z  =  o,  and 
(16)  attains  the  same  values  along  I>,.    The  values  a  =  4,  54<)8<  00  are  reached  for  various 

w's  along  z'  =  wy3  whose  limits  are  a  =  4,  /3= ; , — r .    The  A  value  m=—-hf  being  a 

^  -^  w(27w+4) 

turning-point  for  /3's  denominator. 

(9),  (10)  behave  in  a  similar  manner  and  also  fall  on  the  upper  cusp  region  in  (a,  /3). 

(11),  (14)  are  divided  by  A,  reach  a  =  4  |8=  00  but  /3<o  throughout.  They  fold  on  D\ 
hyperbohc  branch  in  the  second  quadrant. 

(12),  (13)  similarly  fold  on  D  in  the  third  quadrant.  They  reach  a=  00 ,  |8=  co  at  F  and  0 
respectively  and  so  fall  on  the  negative  side  of  B.    They  reach  a  =  o,^  =  o  with  (7),  (8). 

Of  the  sheets  I,  II,  .  .  XII  with  real  iy,z)  but  complex  {a,b,c)  which  all  lie  between  a  =  o 
and  a  =  4; 

I  reaches  a  =  4,  ;8<o  along  z''  =  my^  for  m<  —^S,  reaches  j8=oo  ,  o<a<4  at  0  alongz  =  W3»» 
for  m>l,  reaches  a  =  o,  o>^> — 00  at  o  along  y^  =  4^z',  reaches  /3  =  o,  o<a<4  along  z  =  my' 
for  2='»,  o<w<  I,  and  thus  covers  the  rectangle  o<a<4,  /8<o. 

II  and  III  are  continuous  over  y  =  o  which  is  not  discriminantal.  They  reach  a  =  o, 
o</8<  00  at  o  along  y^=4fiz'.  The  values  are  here  paired  with  those  in  (8)  since  Di  is  closer 
to  the  axis  than  any  of  the  parabolas.  The  sheet  reaches  ^  =  0,  o<a<i  for  z  =  my'',  2=00. 
It  does  not  reach  a  >  i  but  is  folded  with  IV  along  T. 

IV  reaches  /3=oo,  o<a<4  along  QZ:  a  =  o,  o</8<oo  at  Q  on  cubic  parabolas;  joins 
VII  along  j8=oo,o<a<4;  reaches  ;8  =  o,  o<a<i  ioi  z  =  my^  pairing  with  II,  III. 

V  covers  the  whole  rectangle  P<o,  o<a<4.  It  reaches  a  =  o,  /3<o  at  Q:  <x  =  4,  P<o  on 
semicubical  parabolas  at  infinity:  o<a<4,  /3  =  o  for  z  =  my^  atz=<»  :o<a<4,  ^=00  along 
4y— 8z  — I  =0  from  Q  to  00  ,  a  increasing  monotonously. 

VI  reaches  a  =  o,  /3>o  at  Q  the  cubical  parabolas  pairing  the  values  with  IV;  a  =  4) 
o</3<54at^:  o<a<4,/8=oo  with  V  along  4))  — 8z  — 1=0.  It  contains  I>2  and  7  which  goes 
to  the  cusp  and  is  folded  with  VIII  along  D. 

VII  covers  the  rectangle  ^<o,  o<a<4.  It  reaches  a  =  4,  o>^>  — 00  at  A:  a=o, 
o>/3>  — 00  a.tQ:  /8=  00  ,  o<a<  4  along  QZ  joining  IV;  and /8  =  o,  o<o<4  at  .4. 

VIII  covers  the  same  region  as  VI  folding  symmetrically  on  D".  _ 
DC  contains  T  and  ^<o.    A  (0  =  0,  i3  =  o)  is  not  reached.     It  reaches  a  =  o,  /3=  —  ^  on  T 

aXF:  a  =  4,l3<oonz'—my^ioTZ=M,m>o.    j8=a3,  o<a<4  is  reached  with  VIII  along  z  =  o 
f rom  F  to  y  =  =0  :  and  ;8  =  o,  i<a<4  on  z  =  w)''forz=oo,  wj<o. 


THE   TRANSFORMATIONS  79 

X  and  XI  cover  the  same  region  as  IX  folding  on  T. 

XII  covers  the  negative  rectangle.  It  reaches  a  =  o,  /3  any,  at  F  joining  (12) :  0  =  4,  /3  any, 
with  VII  at  ^ :  /3  =  o,  o< a<  4  with  VI  at  A  and  j8=  00 ,  o< a<  4  at  F  for  y  =  inx',  m  >\. 

The  nodal  curve  P  =  o  does  not  affect  the  reaUty  of  the  roots  but  causes  the  sheets  to  cross. 
It  has  asymptotes  0  =  4  .  0  =  9,  /3=27(— 2±i  3).  There  are  no  real  points  for  i<a<3,  nor 
for  \\<^<2'j.  At  1  =  3,  /3=27  is  a  cusp  which  falls  on  the  cusp  of  D  with  coincident  tangent, 
but  the  P  curve  includes  the  D  curve  up  to  a  =  4,  /3  =  54  where  they  cross,  and  up  to  0  =  4,  ;8=  00 
where  they  have  contact.  The  curve  P  meets  T  only  at  a=i,  /3  =  o  and  two  complex  points 
(i4±i   ^)     «_     27(9  + v^^) 


0  = 


,  j8= '^ ,  the  positive  signs  corresponding.    P  and  D  meet  at  the 

22  83 

cusp  and  at  0  =  4,  j8=  00  and  also  at  the  origin  where  the  tangent  to  P  is  30—2/8  =  0. 

The  curves  have  also  two  intersections  for  a=  00  ,  ^=  —  V. 

Since  y  is  given  as  a  rational  function  of  (a,  13)  it  is  real  for  all  real  points,  but  the  z's  may 
form  a  complex  pair.    The  condition  is 

12(1— a)(4)/»— y)+a(6y— 1)'<0. 

This  is  satisfied  for  the  branch  which  reaches  the  origin  in  the  third  quadrant  and  for  the 
branch  between  T  and  D  in  the  second  quadrant,  and  in  the  region  round  the  cusp  of  D  up  to 
the  crossings. 

To  determine  which  compartments  of  the  y,  z  plane  are  paired  by  P  we  consider  first  the 
region  inside  the  cusp  of  D  with  a>4.    The  sheets  involved  are  (i),  (2),  (9),  (10),  (15),  (16). 

At  j8=  00  for  positive  approach  the  roots  are  o  •  j,  J,  ,  _  , ,  — -,  _  . —  . 

Taking  a  =  6  as  a  typical  case  where  P  acts  they  are 

o,  i,  i, -^,  ±-g-. 

Now  (i),  (2),  (9)  can  all  reach  y  =  i  but  (2)  is  the  only  region  reaching  y>l,  y8=  <»  .  Hence 
(2)  has  the  root  —r-  and  (i)  =  (9)  has  J.     (16)  reaches  y  =  o  but  (15)  cannot.     (10)  and  (15) 

o 

have  then  the  negative  values.  At  the  point  a  =  6  on  D  =  o  we  have  (i)  =  (2),  (9)  =  (10),  (16)  = 
(15)  and  as  only  one  crossing  occurs  between  this  value  of  /3  and  /8=oo  it  must  be  (10),  (16) 
which  cross. 

The  cross-section  of  the  surface  has  the  arrangement  of  the  diagram  (Fig.  22). 

Passing  across  /3=  00  (4)  and  (11)  are  paired  by  P  for  a>g. 

Next  consider  the  region  /3<o,  o<a<  i.  Where  P  crosses  a  =  o  three  roots  are  equal  to  j. 
These  are  in  X,  IX,  XII  since  VIII  has  l3>o.  P  can  only  pair  IX  and  XII  since  the  y's  must 
be  the  same  and  a  =  o  for  P  gives  y  =  z- 

The  region  /3>o,  o<a<  i.  Q  approached  on  43;— 8z— 1=0  has  )8=oo  ,  a  =  o.  For  small 
o's  we  have  roots  in  IV,  VI  and  y8>o.  IV  can  only  reach  fi=o  by  moving' to  infinity  along 
z  =  my',  while  VI  must  reach  A.  The  y,  z  diagram  (Fig.  20)  shows  that  this  entails  the  crossing 
of  the  projections  on  the  y-axis. 

The  branches  of  P  at  the  cusp  join  complex  z's.  Using  the  continuity  of  the  real  surface  as 
a  guide  we  see  that  the  sheets  (10),  (16)  crossing  on  P  for  a  >4  represent  two  roots  which  become 
equal  and  infinite  at  0=4  and  then  complex,  and  so  the  nodal  line  is  as  usual  continued  as  a 


<*.    <.   0 


oc  =3 


I    Co  n  II* 


it   Nod^l  Points 


Infinite    At>t^^oacltf^    5^»jmi/oIizCii  ' 


3.         Y 


gfc 


Fov  3i.  =  o    Ihercwti  aw      0  ,  'h  ,'Ih,  '/u  ,  '/i  ,Vs 

^TCclit      ^OV       3  =  0 


0  <<wi  <:  1 


__JL--^ 

01=1 
/    1 

VI    .  - 

VII.   IX 

ivcuiir 

XII — — - 

— ^^ 

(Ol.-t)=-0  PcYiijirimiU  ?o»itii'Yis 


1  «^ot  it  3 


3  ■<•  ot  «:  4 


F«v      Ol  a    ^       tkt    ^0 or.   are     /i     '/j 


4  <  a  <;  p 


oC  >  g 


Fiq  2  2        d  sectiam     of     FCvj  ,  x,  (J>)  t  0 


DETERMINATION   OF   THE   SIDES  8l 

real  isolated  nodal  line  whose  a,  /3  projection  is  the  curve  P  and  whose  y  values  run  from  oo  to 
5  at  the  cusp  and  back  to  <»  at  the  end  of  the  asymptotic  branch. 

The  small  arch  of  P  between  the  origin  and  a=  i,  )8  =  o  pairs  IV  and  VI.  From  the  origin 
along  the  first  vertical  asymptotic  branch  (12)  and  (13)  are  paired;  crossing  a  =00  the  sheets 
are  (11),  (14)  and  the  nodal  line  is  isolated  in  both  these  parts. 

From  0=1,  /3=o  moving  to  /8<  o  the  paired  sheets  are  IX,  XII  until  a  =  0  is  crossed,  when 
(6),  (12)  take  their  places.  Passing  a=oo  (4),  (u)  are  paired.  For  the  last  two  parts  the 
nodal  line  is  ordinary. 

In  a  similar  way  the  equalities  further  marked  in  the  diagram  may  be  established.  The 
result  though  a  compendious  collection  of  information  is  not,  as  remarked  in  case  I,  a  complete 
and  consistent  statement  of  all  the  facts.  In  particular  as  special  defects  the  representation  of 
the  points  a  =  0,  j8  =  o  :  a=i,j8  =  o  :  a  =  4,  j8=  00  ,  for  which  the  equation  becomes  indeterminate, 
and  the  surface  has  a  line  parallel  to  the  y  axis,  is  omitted. 

A  set  of  diagrammatic  cross-sections  of  the  surface  F(y,a,^)  —  o  is  given  in  Fig.  22. 

II.      THE   DETERMINATION    OF   THE    SIDES 

The  side  of  a  triangle  with  given  bisectors  is  a  root  of  a  sextic  equation  whose  coefficients 
are  rational  and  unsymmetric  in  the  bisectors. 
To  determine  these  sides  we  write 

a—b  =  \a,b—c=Ka 
whence 

6  =  a(i  — A,),  c=a(i  — A.— k)  (30) 

The  fundamental  formulas  for  external  bisectors  then  give 

---.=<i^i^-  (3.) 

and  ii  k=  pK 

p(i-2pK-K)=^{i-pK){i-K){i+py  (33) 

qil~2pK-K)={l-pK-K){l  +  K)p^  {^^') 

From  these  quadratics  «  can  be  eliminated  and  we  have 

p'(p+iy+pP<P+i)-qpip+iy-p'p'+pqp{p+i)+q'{p+iy  =  o,  (34) 

and  K  is  given  rationally  in  p  by 

pp-q{p+i)-p(p+j)_ 


pp+q{p+i)-p{p+i){2p+i) 


(35) 


The  explicit  sextic  for  k  has  coefficients  of  the  third  order  in  p,  q  and  only  five  coefficients 
vanish.    It  is  not  as  convenient  for  computation  as  the  chain  above. 

Since  only  the  ratios  of  bisectors  are  used  in  this  we  may  choose  the  scale  so  that  {b—c)=  i. 
The  isosceles  triangles  do  not  enter  in  the  equation  so  that  this  is  permissible.     We  then  have 

I 

a=~  . 

K 

As  in  case  I  we  may  conclude  that  a  is  the  root  of  an  irreducible  sextic  of  the  symmetric 
groupG„„(I,  §  15), 


82  THE  PROBLEM   OF   THE  ANGLE-BISECTORS 

12.      THE    CASE    OF   EQUAL   BISECTORS    (EXTERNAl) 

The  group-theory  argument  of  I,  §  20  may  be  repeated  in  this  case  with  the  same  result. 
The  sextic  for  the  side  a  remains  a  sextic  when  K^L  and  then  gives  three  a's  and  three  6's. 
The  sextic  for  c  reduces  to  the  square  of  a  cubic  but  the  rational  relation  fails  and  it  is  necessary 
to  solve  a  quadratic  also. 

The  explicit  determination  of  a  which  involves  the  solution  of  a  sextic  solvable  by  a  chain 
of  a  cubic  and  quadratic  is  conveniently  performed  by  a  special  method. 

If  k  —  l=i  and  m  be  taken  as  the  single  parameter  of  the  problem  and  we  write  a+b=t 
ab  =  u  the  fundamental  equations  (11)  give  easily 

{a-by  m{t+a-iy .   m{t+b-iY 


{i-2a.){i-2b)ab     i2t-i)(i-t){i-2a)a     (2/- i)(i-/)  (1-26)6 


(37) 


Equating  the  first  fraction  to  the  arithmetic  mean  of  the  second  and  third 

2{P-4u)ii-t)i2t-i)  =  m[-2l^-4u''+st^+ut-4P+t]  (38) 

and  the  equation  Di{a,b)  =  o  is 

2<3-l-4M/-5/2-3M-f4/-i  =  o  (39) 

From  these  we  have  the  cubic 

mit-i){2t-iy=(4t-3)(4i'-5t+2)  (40) 

giving  t,  while  (39)  gives  «  as  a  rational  function  of  t  and  the  sides  are  obtained  by  the  quadratic 
and  linear  equations 

a-|-6  =  /,  a6  =  M,  and  c=i  — a— 6. 


Ill 

I.      THE    MIXED   PROBLEM 

Given  two  bisectors  at  one  vertex  and  another. 

In  this  case  the  symmetry  being  destroyed  a  special  unsymmetric  method  is  usfed.  The 
cases  where  the  third  bisector  is  internal  or  external  are  reached  by  a  change  of  sign  in  a  side 
and  can  be  treated  in  one  set  of  equations. 

As  data  we  take 

If  A,  B,  C,  are  the  angles  of  the  triangle  the  fundamental  formulas  give 

*=tan ,  o=sm  B  cos ^sm  A  cos .  (2) 

2        ^  2  2 

If  the  angles  are  determined  the  remainder  of  the  problem  is  an  affair  of  ruler  and  compass. 
The  problem  of  determining  the  trigonometric  functions  of  the  angles  involves  an  irration- 
ality of  the  tenth  degree,  the  root  of  an  equation  whose  group  is  the  symmetric  group. 
We  write 

B-C=6,  B=2M,qcos-  =  H,  (3) 

and  treating  H,  9  as  assigned  M  as  required 

H  sin(43/-(9)-sin  2M  •  sin  {2,M-e)  =  o.  (4) 

For  the  purpose  of  determining  the  group  we  write 

e'"=x,  2Hi=k,  e-"^=h, 

and  obtain  the  algebraic  equation 

kx{ho^—  i)—{x*—  i){hx^—  i)  =  o,  which  is  obviously  irreducible.  (5) 

2.      THE    MONODROMIE    GROUP 

For  h  —  osix  roots  of  the  equation  are  infinite,  the  other  four  being  given  by 

x^—kx~i  =  o. 

This  has  double  roots  for  four  values  of  k,  y. 

44 

k^= ,  T,x^-\-i  =  o,  k  =  4x'. 

27 

For  k  =  o  the  roots  are  +1,  +i,  —i,  —i  and  may  be  named  from  this  position  in  this  order 

(7),  (8),  (9),  (o). 

83 


84  THE  PROBLEM   OF   THE  ANGLE-BISECTORS 

If  k  =  pe and  p  decrease  from  o  to  -, — tj  the  roots  (7),  (8)  approach  and  (9),  (o)  depart 

from  the  origin  in  the  y  plane.     The  path  is  for  y  =  re'^. 

^•(1  +  2  sin  26)=  I 
in  rectangular  co-ordinates 

{x^+y^)  {x^+4xy+y'')  =  i . 

At  the  extreme  value  mentioned  for  p,  (7)  and  (8)  are  equal  and  give  rise  to  a  two-cycle. 
By  moving  k  along  th«^  same  radius  to  the  corresponding  positive  value  of  p,  (9)  and  (o)  become 
equal. 

By  using  the  perpendicular  radius  as  a  path  for  k,  (o),  (8)  and  (9),  (7)  respectively  become 
equal  in  the  two  directions,  the  path  being 

r*(i  —  2  sin  26)  =  I. 

Hence  we  get  all  interchanges  on  (7),  (8),  (9),  (o). 

Returning  to  the  six  infinite  roots  we  find  that  they  form  a  six-cycle  iorh  =  o  for  all  finite  k's. 

For  ^=cc  of  (7),  (8),  (9),  (o)  three  become  infinite  and  one  approaches  zero.  This  is 
obviously  the  negative  root  of  x*—kx— 1  =  0  and  was  named  (o)  in  the  original  position. 

For  the  approach  ^  =  00  followed  by  A  =  o  nine  of  the  roots  are  infinite  and  eight  form  a 
cycle.     Since  every  interchange  of  (7),  (8),  (9),  (o)  is  allowed  we  take  this  cycle  as 

[(i),  (2),  (3),  (4),  (S)>  (6),  (7).  (8)]  in  some  order. 

Transforming  [(o),  (7)]  by  the  powers  of  this  substitution  we  have  with  [(o),  (9)]  every 
[(o),  («)]  and  so  the  symmetric  group. 

3.      THE    EQUATION   FOR    THE    TANGENT    OF    A   HALF-ANGLE 

Writing  y=  tan  M,  t=  tan  0,  and  k=2Hi  the  equation  (4)  becomes 

H'{y'+i)(,ty^+4f-6yH-4y+ty-4y'iy3-syH-sy+ty  =  o  (6) 

The  coefficients  are  rational  in  {k,  h,  i)  and  y  is  rational  in  {x,  i).  The  group  is  also  the 
symmetric  group. 

For  every  y  there  are  two  x's  but  ten  belong  to  F{x,  —k,h)  =  o.  These  in  the  real  cases 
correspond  to  values  of  M  increased  by  ir,  and  lead  to  the  same  triangles.  So  for  a  change  of 
sign  mp,t  and  y  change  also  and  the  same  triangles  occur,  and  without  loss  of  generality  we  may 
take  p,  q  as  positive. 

By  differentiation  and  elimination  of  H'  we  obtain  as  a  discriminantal  equation 

/y9-6//^-  i5/y+  (/'-  2o)/+2i>'5-  2i/^y4-  2i/y+  (3/2-  i2)y^-\-()y-  1=0  (7) 

Treated  as  a  quadratic  in  t  the  discriminant  is 

(>''+i)V-36/-i2). 

Hence  real  roots  only  occur  for  |  y  |  <  6 .  02  .  .  . 

The  real  discriminantal  curve  is  then  determined  rationally  in  y  and  a  square  root  of  a 
function  of  y.  It  may  be  traced  by  assigning  to  y  all  real  values  outside  the  critical  values  and 
calculating  I,  H  and  so  p,  q.  The  curve  has  quadrantal  symmetry  in  the  {p,  q)  plane.  The 
critical  values  correspond  to  p=  .  1456  .  .  q=  .  2851  ... 


EQUATION  FOR  TANGENT  OF  A  HALF-ANGLE 


8S 


From  this  value  t=  2974.  .  one  value  of  t  increases  monotonously  to  00  and  the  other 
decreases  monotonously  to  zero.  The  p,  q  values  tend  each  monotonously  to  (1,0)  and  (o,  ^) 
respectively. 

.  There  is  a  cusp  at  (o,  5)  where  <  oc  -,/>«:  — ,  and  q—^  «:  ~  .    At    (i,  o)    the   proper 

approximation  is  a  pair  of  parabolas. 


FltZ3 


Under  [lermutiitions  o\  (^,<J.K) 
tht|30(nT5  1.2,5  corT«s)3ont( 
-3.(|i. 3, C[=  10)  lias   Svfiil.Yoots 


The  curve  is  continued  past  p=  i  by  negative  values  of  y  and  t,  which  double  change  of 
sign  leaves  H'  unchanged  while  (/>,  q)  are  transformed  to  ( -  ,  |  j,  the  same  transformation  as 

that  effected  by  the  interchange  of  two  of  the  assigned  bisectors,  or  say  of  (/,  g)  where 

III 
I  :  p  :  q  :  :  ~  :  -  :  t. 
f     g     ft 

In  addition  to  the  real  branches  thus  traced  there  is  a  conjugate  point  ^=  00  ,9  =  0  (Fig.  23). 


86 


The  graph  of 


THE   PROBLEM    OF    THE   ANGLE-BISECTORS 
4.      THE   SOLUTION    OF   THE   SEAL   PROBLEM 

6 

H  =  q  cos  -  =  sin  2M  sin  {i,M—0)-^sm  {4M—O) 


is  of  the  same  general  character  for  any  0.    The  diagram  is  for  0=  -  (Fig.  24). 

4 

In  general  the  zeros  are  :  o,  -  ,  tt  -  tt  independent  of  d,  and  — | ,  m=i,  2,  .  .0. 

22  33 


The  infinities  are  — | ,  m=i,  2, 

4      4 


8. 


r.^u 


XV  h 


As  p  ranges  from  o  to  co  ,  0  ranges  from  o  to  ir,  and  these  values  coalesce  only  for  6= 
f\ 

-  when -  +  -=- .     This   corresponds   to   p=i  and   two   zeros   coalesce  but  remain   real  on 
2332 

passing  the  value,  (3)  and  (4)  being  interchanged. 

For  given  H  the  roots  are  either  10  real  or  8  real.     The  critical  values  can  be  found  from 

the  derivative  vanishing  at  the  roots  of 

2  cot  2^—4  cot  (43/— ^) +3  cot  (3JW— ^)  =  o. 
These  values  can  be  found  without  much  trouble  from  the  table  of  natural  cotangents. 


CHARACTER   OF   THE   SOLUTIONS  87 

The  values  of  the  roots  which  are  then  entirely  separated  can  be  found  either  by  Horner's 
method  from  the  equation  (6)  or  by  trial  from  the  table  of  logarithmic  sines,  and 

log  H=log  sin  2Af+log  sin  (33/— ^)  — log  sin  {4M—O)  (8) 

The  triangles  are  all  real  and  possible  il  y  or  M  is  real,  for  in  all  cases  A  +  B+C=Tr.  The 
values  oi  A,B,C  are  not  always  positive  and  the  results  are  subject  to  an  interpretation  by 
way  of  interchanging  internal  and  external  bisectors. 

5.      THE   CHARACTER   OF    THE    SOLUTIONS 

First  take  o</'<  I,  i.e.  o<0<-  . 

'^  2 

For  the  root  (i) 

o<M<^  :  o<B<-  :  -e<C<-^  :  -e<B+C<o. 
42  2 

Since  sin  yl=sin  {B+C)  we  have  fl<o,  6>o,  c<o  and  the  solution  refers  to  K,  K,  L  in 
place  of  K,  K,  L. 
For  the  root  (2) 

'-<M<'V-  :  ^-^5<-V  :  -'<C<--'  :  '<B+C<.. 
3  4     4«     3  223  223 

Hence  a>o,6>o,  and  c  is  of  doubtful  sign.     Fot  q<pjj<o. 

So  for  q<p  (2)  gives  a,b,c  :  +,  +,—  and  refers  to  K,  K,  L  and  for  q>p,  a,b,c  :  +,  +, 
+  and  refers  to  K,  K  L  that  is  to  the  original  verbal  statement  of  the  problem^  _ 
The  roots  (3),  (4)  are  alike  and  have  a,  b,  c  :  — ,  +,  +  and  refer  to  K,  K,  L.      . 
For  the  root  (5) 

e    ,    TT         ,,        d   ,2Tt 

'+-<M<-^ . 

42  3      3 

Here  a  is  always  positive,  b  negative,  and  c  changes  sign  when  C  =  tt  or  when  ?=  i. 
The  other  cases  are  of  invariable  class  and  the  results  may  be  collected: 
I.     The  original  case  a,b,  c  :  -\- ,  +,  -{-  occurs  for  (6)  and  (2)  if  q>p. 
II.     The  case  a,b,  c  :  -\-,  - ,  -  occurs  for  (3),  (4),  (8),  (10)  and  (5)  if  q<x. 

III.  The  case  a,b,c  :  +,  -,  +  occurs  for  (5)  i{  q>  1  and  for  (i). 

IV.  The  cased,  6,  c  :  +,  +,  -  occurs  for  (7),  (9). 

The  cases  where  p>i  can  be  included  by  noticing  the  transformation  (/'»  t)  combined 

with  iq,  -)  which  entails  (B,Tr—6)  and  if  also  we  interchange  (Af,  ir—M)  the  original  equa- 
tion is  unchanged. 

This  interchange  however  takes  sin  B  to  —  sin  B  and  leaves  sin  yl,  sin  C  invariant.  The 
triangles  are  unchanged  but  the  internal  and  external  bisectors  at  A  are  interchanged.  The 
classes  of  solutions  (I,  III)  and  (II,  IV)  are  interchanged  in  the  pairing  given. 


88  THE   PROBLEM   OF   THE   ANGLE-BISECTORS 

_     The  whole  transformation  is  equivalent  to  an  interchange  of  the  fundamental  quantities  K, 
K,oriiK:K:L::f:g:h  to  {f,g). 

Under  (g,  h)  which  involves  {p,  q),  however,  a  new  problem  arises,  and  so  under  (/,  h) 

which  replaces  *  by  -  and  o  by  -  . 

q  q 

Of  the  six  permutations  of  (/,  g,  h)  three  sets  of  two  lead  to  distinct  sets  of  triangles. 
Namely  for  this  problem 

(/,  g,h)  =  {g,f,h)  corresponds  to         {p,q)  =  (^,^  . 

ih,f,g)=(f,h,e)  (q,p)  =  (^i,t'j. 

6.      THE    CASE    or   EQUAL   BISECTORS 

For  p  =  q=i  the  equation  for  a:=tan  —  becomes 

{x'+i){x*-6x^+i)-Sx'(sx^-iy  =  o, 
which  reduces  to 

(x^— i)(x''— i4r'-|-i)(a;''-(-4a;^-|-i)  =  o  (9) 

The  first  factor  gives  in  two  ways  the  triangle  A=B  =  - ,C  =  o. 

The  second  factor  gives  in  four  ways  the  triangle  ^=5  =  -,C=  —  . 

The  third  factor  gives  in  four  ways  the  triangle  determined  by 

B=2  tan-'I/(i/5-2) 

or  approximately  when  the  angles  are  taken  positively  and  internal, 

^  =  13°  40',  5=128°  10',  C  =  3&°  10'. 

In  this  case  the  B  bisector  is  external,  in  the  second  case  it  is  internal,  and  in  the  first  the 
words  internal  and  external  have  no  proper  distinction.  The  case  oi  p=i,q  any,  has  an  equa- 
tion containing  only  even  powers  of  x.  If  the  problem  were  solved  in  terms  of  the  sides  the 
locus  p=i  would  be  discriminantal,  but  for  this  equation,  although  the  group  reduces  so  that 
the  equation  may  be  solved  by  solving  a  quintic  and  quadratic,  there  are  no  equal  roots,  the 
roots  merely  referring  to  the  same  five  triangles  in  pairs.  In  the  former  cases  we  had  the 
phenomenon  of  a  discriminantal  locus  in  one  solution  corresponding  to  a  locus  of  reducibility 
for  another;  here  we  have  it  corresponding  to  a  locus  of  group  reduction. 


IV 


I.   THE  GENERAL  PROBLEM  FOR  REAL  DATA 


If  three  real  numbers  are  assigned  as  the  lengths  of  any  three  bisectors  the  problem  of  deter- 
mining the  triangle  is  to  be  solved  by  successive  application  of  the  methods  of  the  three  cases. 
The  number  of  real  solutions  depends  on  the  data,  and  the  character  of  the  dependence  is 
revealed  by  considering  the  discriminants  of  the  three  cases  simultaneously. 

In  cases  I  and  II  any  three  assigned  real  numbers  cause  the  (a,  /3)  point  to  fall  in  the  region 
within  the  cusp  of  D{a,  fi).  For  a>4  I  has  8  real  solutions  with  possible  triangles;  II  has  4. 
For  a<4  the  numbers  are  7  and  o  respectively. 

The  condition  0  =  4  is  expressed  in  (/>,  q)  as  the  vanishing  of  the  product 

{p+q+l){p^q-l){p-q+i){~p+q+l). 

On  account  of  the  symmetry  of  the  discriminant  of  III  it  is  only  necessary  to  consider  one 
quadrant  of  the  (/>,  q)  plane  (Fig.  24). 

Taking  the  first  quadrant  for  p+q—  i  >o  we  have  a<4.  There  are  then  three  regions  and 
three  classes  of  the  general  problem: 

Class  A  :  a<4  and  A3<o. 
Class  B:  a>4  and  ^i<o 
Class  C:  o>4  and  A3>o 

For  class  A,  I  has  7,  II  has  o,  and  III  has  3  permutations,  each  of  which  has  8  solutions. 
The  permutations  of  if,g,h)  leave  the  square  (0  =  4)  invariant. 

The  total  for  class  A  is  then  33  real  solutions  with  proper  triangles.  For  class  B  the  per- 
mutations of  (/,  g,  h)  do  not  carry  the  representative  point  across  the  discriminantal  curve. 
Ill  has  then  three  sets  of  8  solutions,  I  has  8,  and  II  has  4,  the  total  being  36.  For  class  C  q<^ 
and  its  reciprocal  occurring  as  a  9  under  {h,f)  is  outside  A3.    The  other  transform  is  inside  Aj. 

For  this  class  two  sets  in  III  have  10  real  solutions  and  I  has  8,  and  II  4  solutions  :  in 
all,  40.     This  is  the  greatest  number  and  occurs,  for  example,  if  /  :  g  :  /;  : :  3  :  30  :  10  (Fig.  23). 

This  case  has  been  taken  for  the  triangles  in  the  illustration  (Figs.  25,  26,  27). 


2.      THE    PROBLEM   WHEN   A   RIGHT   ANGLE    AND   TWO   BISECTORS   ARE    GIVEN 

Taking  the  right  angle  as  C  the  sides  b  and  c  are  rational  functions  of  the  side  a. 

2a— I  —  2a^-f2a— I  ,  ,  , 

b  =  —, -s,    c= -. r — ,    a+o+c=i 

2(a— i)  2(a— i) 


(i) 


By  interchange  of  (a,  b)  and  by  changing  the  signs  of  sides  the  fifteen  pairs  of  the  six  bisec- 
tors can  be  reduced  to  three  cases. 

89 


WHEN   A   RIGHT  ANGLE   AND   TWO   BISECTORS   ARE   GIVEN  91 

Case  I.    Given  K,L. 

The  ratio  -^^  becomes  a  perfect  square  in  (a,  j   2)  namely 

-=;=2]/2a(a— 1)^-^(1  — 2a)^  (2) 

If  Z,  H-  2  j/2  A"  is  plotted  against  a  the  same  curve  which  occurred  in  the  internal  problem  (I,  §  Sj 
is  given. 

For  -p>o  there  is  then  one  real  solution  of  the  cubic.     This  gives  a  real  triangle  with  all 
K. 

positive  sides  and  is  the  solution  of  the  problem  as  stated,  namely  K,  L  are  internal  bisectors. 

For  t;<o  there  are  three  solutions: 
K 

i)  a>i,  b>i,  c<o.    The  range  of  c  has  a  maximum  at  c=  — |   2— i  corresponding  to 

the  right-angled  isosceles  triangle  and  c  tends  to  —  00  as  either  a  or  6  tends  to  <x> . 

The  bisectors  are  both  external. 

2)  i>a>5,  —  <»  >b>o,  <x>  >c>^.     The  A  bisector  is  external,  the  B  bisector  internal. 

3)  o>a>— 00,  5<6<i,  5<c<oo.     The  ^  bisector  internal,  the  5  bisector  external. 

In  all  there  is  a  single  solution  iot  each  arrangement  of  the  bisectors  as  internal  or  external. 
The  discriminant  of  the  cubic  is 

L{8i/2L'+i3LK+8i/2K')  ,  . 

The  double  points  are  at  L  =  o,  ^=0  and  t^  =  ^~^^    ,^^}-    ^     . 

Case  II.    Given  K,M. 

The  equation  for  the  side  a  is  a  sextic;  obviously  irreducible. 

SA'Ca-  iya'-M'{2a''- 1)^(20'-  20+ 1)  =  0  (4) 

8K^ 
Plotting -jrTj=/>  against  a  we  have  the  real  graph  (Fig.  28).     At^  =  o  a  double  transposition  can 

be  effected  (1,2)  (3,4)  and  at  /»=  00  a  two-cycle  and  a  four-cycle  which  must  separate  5,  6  which 
are  conjugate  complex  for  the  real  approach.  It  may  properly  be  denoted  by  (2,3)  (1,5,4,6). 
Approaching  /)=  00  from  the  negative  side  no  double  point  is  encountered  between  p  =  o  and 
/)=  —  CO  since  the  double  points  are  at  p  =  o,  00  and  the  four  complex  roots  of  the  remaining 
factor  of  the  discriminant : 

i6/)4-  i52/)3-|-93/)^-f-5i2/)+32768. 

The  corresponding  values  of  a  being  given  by 

6a''—  8a3+  'ja'—4a+ 1  =  0. 

Since  the  conjugate  pairing  must  be  kept  the  cycle  at  /»=  °o  must  be  for  this  approach  either 
(1,2)  (3,5,4,6)  or  (3,4)  (1,5,2,6)  or  (5,6)  (1,3,2,4). 

Since  the  complex  double  points  are  distinct  and  the  second  derivative  does  not  vanish  at 
them,  a  single  transposition  occurs  as  an  element  of  the  monodromie  group.  It  is  then  easy 
to  see  that  the  monodromie  group  and  therefore  the  algebraic  group  is  the  symmetric  group, 


It 


FH.Z6 


SPECIAL   CASES   OF   ISOSCELES   TRIANGLES  93 

however  the  cycle  for  />=  —  <»  be  named.     The  discriminant  not  being  a  square,  adjunction 
of  \/p  does  not  reduce  the  group. 

To  discuss  the  character  of  the  solutions  we  divide  into  classes  by  /)<8  that  is  A'<M  and 
by  the  values  for  a  when  p  =  8. 

Class  la.      K<M,a< =,  i>b>—-^,  c<i.    Af  is  external.  ^ 

y  2  1    2 

ClassIIa.     K<M,  -.2516.  .  >a> ^,  .6005  .  .  <6<— -,    .6511,    <c<i.     M  is 

]/  2  ]    2 

external  and  the  general  character  of  the  figure  is  the  same  as  in  la. 

Class  Ilia.     K<M,   -^>a>h ^<6<o,  i>c>^.    A' and  M  are  both  external. 

V2  )    2 

Class  IVa.     K<M,   -^<o<  .7600  ,.,  --^>6>-i  .083  ..,  i<c<i. 323  ..  ,Aand 
V  2  )    2 

M  both  external. 

Class  16.     A:>Af,  —  .2Si6<a<o,  .6oo5>J>i,  .65ii>c>i.     Af  is  external,  A"  internal. 

Class  lib.  K>M,  o<a<^,  ^>b>o,  c  has  the  value  \  at  each  end  of  the  range  and 
decreases  from  these  values  each  way  to  a  minimum  1  2—  i  which  belongs  to  the  right-angled 
isosceles  triangle. 

K  and  M  are  both  internal  and  this  case  is  the  only  one  solving  the  problem  verbally 
expressed  for  internal  bisectors. 

Class  III6.  K>M,  .7600.  .  <a<i, -i  .o83>6>- <»  ,  1323.  .  <c<oo.  Both  bisec- 
tors are  external. 

Class  INb.  K>M,  i<a<oo,  co  >b>i,  c  has  the  value  —  00  at  each  end  of  the  range 
and  reaches  a  maximum  —  |    2—  i  for  the  case  of  the  isosceles  triangle.     K  is  external. 

The  approximate  values  for  a  are  the  roots  of  the  equation  for  equal  bisectors.  For  this 
case  only  two  real  non-trivial  solutions  exist : 

a  :  b  :  c  ::  —  .2516  .  .  :  .6005  .  .  :  .6511  .  .  The  angle  A  about  22°4o',  K  internal,  M 
external. 

a:  b  :  c  ::  .7600  .  .  :  —1.083.  >  '■  1-323  •  ■  ,  A  about  3S°4',  K  and  M  both  external. 
Case  III.    Given  two  bisectors  at  one  vertex  and  the  right  angle. 

The  tangent  of  half  the  difference  of  two  angles,  and  one  angle  are  given,  hence  the  prob- 
lem is  one  for  ruler  and  compass. 


3.      SPECIAL   CASES    OF    ISOSCELES   TRIANGLES 

The  general  method  of  Case  II  leaves  the  construction  of  an  isosceles  triangle  given  an 
external  bisector  at  the  base  indeterminate. 
Other  conditions  must  be  given. 
If  the  base  a  and  the  external  bisector  L  are  given  we  have 

j,_{-a+2b)a'b 

^-    (b-ay    • 

To  determine  the  angles  write  ==  p  =  -. — ^  where  <^  is  half  the  external  angle  at  the  base. 
^  X  sm  2</) 


I.  CIO 


IDENTICAL   RELATIONS   AMONG   THE   SIX   BISECTORS 


95 


The  solution  is 


cos  <i>- 


/>±1    (j)'+4) 


The  problem  can  be  solved  by  ruler  and  compass  and  is  an  extension  of  Euclid's  decagon 
problem  (/>=  —  i). 

The  sign  of  p  does  not  determine  any  representable  difference  of  configuration,  but  for 

^  <  I  /)  I  <  -  one  triangle  has  the  bisector  internal  :  below  the  lower  value  two  solutions 

V/'2  2 

have  external  bisectors,  above  the  higher  one  triangle  is  complex. 

If  the  sides  b  =  c  and  L  be  given  the  problem  requires  the  solution  of  a  cubic  equation. 

Namely  \i-j-=K,  and  t  =  ■^, 

X3+  ((C-  2)A^—  2k\-\-k=o. 


F,ff.Z«. 


The  discriminant  is  k(4k^+ 13(0+32),  and  k=<x   is  also  discriminantal. 

For  K>o  there  are   three  real  roots,  one  negative  referring  to  an  internal  bisector  and 

two  positive  referring  to  external  bisectors.     For  k=  i  the  angle  ^  is  —  and  the  bisector  inter- 
nal, or  vl  is  —  or  -  and  the  bisector  external. 

7        7 


4.      THE    IDENTICAL   RELATIONS   AMONG    THE    SIX    BISECTORS 


IT 

I.     We  have  =  =  tan 
K 
whence 


{'~-^)^ 


^iE\^u'^ 


(s) 


II.  The  length  of  the  line  joining  the  extremities  of  the  two  bisectors  from  A  is  r^ 

Hence 

„  I 


2abc 
J' 


y\K'+K') 


-=o 


(6) 


96  THE   PROBLEM   OF    THE   ANGLE-BISECTORS 

III.  The  altitude  of  the  right  triangle  included  by  the  bisectors  at  A  and  the  line  joining 

their  extremities  is 

K  •  K  9 

//a= „  ■  and  as  altitude  of  the  triangle  Ha=  —  , 

2'    {K'+K')  2a 

where  5'  is  the  area  of  the  triangle. 
From  this  equality 


a  =  S  .  UKl  +  K^  0 

K-K  " 


while  K''      s  ■  {s-a){b-cy 


K^     {s-b){s-c){b+cy 


By  substituting  for  a,  b,  c  from  (7)  in  2=^  a  third  relation  is  obtained. 
This  is  for  convenience  expressed  by  writing 

K  K 


and 

when  it  takes  the  form 


jjk'^     {^p)mip-qy 

K     U(p+q-r)nip+qy  ^^^ 


It  is  to  be  noticed  that  p  :  q  :  r  : :  a  :  b  :  c. 

The  independence  of  the  conditions  I  and  II  is  obvious.     For  III  the  set  of  values 

satisfy  I  and  II  but  not  III. 


5.      THE    INDIRECT    PROOF 

In  a  series  of  papers  in  Phil.  Mag.,  IV  (1852),  the  problem  of  the  triangle  with  two  equal 
angle-bisectors  is  made  the  text  (with  some  other  elementary  problems)  of  a  discussion  as  to 
the  necessity  of  the  reductio  ad  absurdum  in  geometry.  The  chief  parts  were  taken  by  Sylves- 
ter and  De  Morgan. 

De  Morgan  claims  to  see  "identity  in  'Every  A  hB  and  every  not  B  is  not  ^"'  by  a  pro- 
cess of  thought  prior  to  syllogism;  and  so  denies  the  necessity  of  an  indirect  proof  in  any  case. 

Sylvester  surmised  that  "The  reductio  ad  absurdum  not  only  is  of  necessity  to  be  employed, 
but  moreover  in  propositions  of  an  affirmative  character,  need  never  be  employed  except  when 
the  analytic  demonstration  is  founded  on  the  impossibility  or  inadmissibility  of  certain  roots 
due  to  the  degree  of  the  equation  implied  in  the  conditions  of  the  question.  If  this  surmise 
turn  out  to  be  correct  we  are  furnished  with  a  universal  criterion  for  determining  when  the  use 
of  the  indirect  method  of  geometrical  proof  should  be  considered  valid  and  admissible  and  when  not." 

It  is  difficult  to  deny  De  Morgan's  general  proposition  though  his  immediate  application 
is  a  little  unfortunate.     The  problem  being,  as  stated  by  Sylvester,  "To  prove  that  if  from  the 


1        >    ■>  1 

1  1  ^ 


THE   INDIRECT  PROOF  97 

middle  of  a  circular  arc  two  chords  be  drawn,  and  the  remoter  segments  of  these  chords  cut  off 
by  the  line  joining  the  end  of  the  arc  be  equal,  the  nearer  segments  are  equal."  The  doubtful 
word  is  of  course  "remoter."  If  this  word  means  every  point  of  which  is  remoter,  then  De 
Morgan's  contention  that  "proving  that  the  inequality  of  the  nearer  segments  makes  the 
inequality  of  the  remoter  ones  follow,  the  equality  of  the  remoter  ones  makes  the  equality  of 
the  nearer  ones  follow  "  is  a  proper  special  case  of  his  general  argument,  can  be  made  good.  This 
interpretation  has  however  a  disadvantage  from  the  geometric  point  of  view.  It  is  not  appli- 
cable to  the  allied  problem  where  the  analytic  geometer  would  say  the  chord  has  complex  points 
of  intersection  with  the  circle.  Yet  for  this  problem  an  entirely  analogous  theorem  is  true  and 
it  is  desirable  to  so  state  the  problem  that  both  cases  are  included. 

If  the  problem  be  stated:  "A  line  of  given  length  has  one  extremity  on  a  straight  line,  the 
other  on  a  circle,  and  the  line  passes  through  a  cut  of  the  circle  and  the  perpendicular  on  the 
given  line  from  the  center  of  the  circle  which  is  not  separated  from  the  foot  by  the  second  cut; 
then  if  the  length  of  the  line  be  less  than  the  distance  from  foot  to  second  cut,  and  in  case  the 
foot  is  outside  the  circle  greater  than  the  mean  proportional  between  twice  the  distance  from 
foot  to  second  cut  and  the  distance  from  foot  to  first  cut,  and  greater  than  the  mean  propor- 
tional between  four  times  the  diameter  of  the  circle  and  the  distance  from  foot  to  first  cut, 
four  positions  are  possible  for  the  line,  and  these  have  symmetry  in  pairs,  and  for  each  sym- 
metrical pair  the  segments  cut  from  the  line  by  the  circle  are  equal" — then  it  is  possible  that 
justice  has  been  done  to  the  facts.  However,  in  the  general  case  the  segments  by  the  circle  are 
neither  nearer  nor  remoter  and  from  the  inequality  of  the  circle  segments  the  inequality  of  the 
circle  line  segments  does  not  follow  without  a  specification  of  the  pairing.  The  syllogist's 
difficulty  lies  in  the  definition  of  the  classes,  and  in  this  special  case  the  class  is  at  least  not 
conveniently  defined  by  equations  alone. 

Turning  to  Sylvester's  view  we  note  that  the  proof  that  equal  internal  bisectors  implies 
isoscelism  falls  very  neatly  in  his  scheme  but  the  corresponding  problem  for  external  bisectors 
presents  a  new  difficulty.  Sylvester  with  the  proper  mathematical  instinct  generalized  the 
problem  before  solving  it:'  namely,  he  said  divide  the  internal  angle  in  a  given  ratio  instead  of 
bisect.  This  generalization  unfortunately  does  not  include  external  bisection.  To  compare 
the  two  cases  we  write  the  equation 

K—L        c        a+6-f-cr   c       b{a-\-b-\-2c)       a 


K+L        b+c 


[_c b{a+b+2c)       a  "I 
a+c         (a+cY        b+c] 


This  holds  from  the  fundamental  equations  for  internal  bisectors,  and  the  spirit  of  Sylvester's 
method  is  to  say — 

The  right-hand  side  is  essentially  positive  for  a-positive  non-trivial  triangle,  and  is  more- 
over expressed  in  products  and  sums  of  products  of  ratios  each  geometrically  interpretable. 
If  then  K  —  L  =  o,  b  —  a  =  o  or  the  axiom  of  Archimedes  fails. 

In  the  external  case,  however, 

K  —  L      a+b—c       c      rc3—{a+b)c''+sabc—abib+ay 


tci-  {a+b)c'+ sabc-  abib+an 
{b-c){c-ay'    '       ] 


b-a         K+L    ib-c)\ 

the  last  factor  in  the  numerator  may  vanish  for  positive  non-trivial  triangles.     It  is  in  fact 

■  Blichfeldt,  Annals  of  Malh^.,  II,  4.22,  gives  the  same  generalization.     His  proof  is  valid  also  for  non- 
Euclidean  space. 


98  THE    PROBLEM   OF   THE   ANGLE-BISECTORS 

—  Diia,  b)  and  the  curve  I>2  =  o  actually  enters  the  region  of  proper  triangles  in  the  (a,  b,  c)  plane 
(Fig.  18). 

In  this  case,  however,  the  non-isosceles  triangles  with  equal  external  bisectors  have  the 
bisectors  oppositely  directed  so  that  further  specification  of  the  conditions  of  the  problem 
which  must  again  presumably  be  by  means  of  inequalities  and  not  equations  will  permit  the 
proof  as  above  by  Archimedes'  axiom. 

It  would  appear  that  in  general,  though  the  difficulties  of  expression  may  be  great,  any 
theorem  true  analytically  for  a  properly  restricted  class  might  conceivably  be  thrown  into  a 
form  similar  to  the  above  and  further  a  direct  geometric  proof  might  be  given  by  Archimedes' 
axiom  and  adequate  restrictions  based  on  order  postulates. 

6.      GENERALIZATIONS    OF    THE   PROBLEM 

Sylvester's  generalization  {loc.  cit.)  which  substitutes  division  of  the  angle  in  a  given  ratio 
for  bisection  does  not  include  the  external  case  as  well  as  the  internal  under  the  same  general 
formulas.  The  same  thing  is  true  if  for  bisectors  which  meet  sides  in  points  dividing  them  in 
the  ratio  of  adjacent  sides  we  substitute  lines  through  the  vertices  dividing  opposite  sides  in  a 
ratio  compounded  of  the  ratio  of  adjacent  sides  and  the  ratio  of  a  corresponding  pair  of  three 
assigned  numbers. 

To  Professor  E.  H.  Moore  is  due  a  generalization  embracing  both  the  internal  and  external 
cases  in  one  set  of  formulas. 

He  introduces  three  parameters  u,  v,  w  and  defines  the  given  quantities  by 

,^j  ,  ,     ,_{au+bv+cw){—au+bv+cw)bcvw 

Ka{u,v,w;  a,b,c)  = ^1— r- 

{bv+cwy 

kI{u,v,w;  a,b,c)  =Ka{u,v,w;  b,c,a) 

Kc{u,v,w;  a,b,c)  =Ka{u,v,w;  c,a,b) 

Then  for  {u,v,'w)=  {1,1,1)  the  internal  formulas  are  given  and  for  {u,v,w)={i,  i,  —  i)  the  exter- 
nal formulas. 

The  problem  for  the  spherical  triangle,  the  formulas  are 

,,     4  sin  s  sin  is— a)  sin  b  sin  c 
^r\^K  =  ^ ■  \i,/  N ,    etc. 

It  may  be  noted  that  the  dual  spherical  problem  reduces  as  the  sphere  becomes  a  plane 
to  a  ruler  and  compass  problem.  Given  the  angles  which  the  medians  make  with  the  sides  to 
construct  the  triangle. 


:  : •••:*• 


VITA 

Richard  Philip  Baker  was  born  February  3,  1866,  at  Condover,  Shropshire,  England,  and 
was  educated  at  Clifton  College,  Bristol  (1877-84),  and  at  Balliol  College,  Oxford  (1884-S7). 
He  graduated  with  the  degree  of  B.Sc.  at  the  University  of  London  in  1887. 

In  1888  leaving  England  for  the  United  States  he  studied  law  and  was  admitted  to  the 
Texas  bar  in  1890.  In  1895  he  became  a  graduate  student  of  mathematics  at  the  University- 
of  Chicago  where  he  attended  courses  by  Professors  Young,  Laves,  Maschke,  Bolza,  Moore, 
Dickson,  and  Wilczynski.  After  several  years  of  teaching  in  secondary  schools  he  became  in 
1905  instructor  in  mathematics  at  the  State  University  of  Iowa.  Here  he  studied  physics  with 
Professors  Guthe  and  Stewart.  In  1910  he  was  appointed  assistant  professor  of  mathematics 
in  the  same  university. 


99 


